Industrial Organization Theory and Applications: Teoria e Aplicações da Organização Industrial por Oz Shy - MIT Press

Page iii Industrial Organization Theory and Applications Oz Shy The MIT Press Cambridge Massachusetts London England Page iv Copyright 1995 Massachusetts Institute of Technology All rights reserved No part of this book may be reproduced in any form by any electronic or mechanical means including photocopying recording or information storage and retrieval without permission in writing from the publisher This book was typeset by the author using the LATEX document preparation software developed by Leslie Lamport a special version of Donald Knuths TEX program and modified by the LATEX3 Project Team All figures are drawn in LATEX using TEXcad by developed by Georg Horn and Jörn Winkelmann The book was complied using emTEX developed by Eberhard Mattes Cameraready copy was produced by Type 2000 Mill Valley California and the book was printed and bound by The MapleVail Book Manufacturing Group Binghamton New York Library of Congress CataloginginPublication Data Shy Oz Industrial organization theory and applications Oz Shy p cm Includes bibliographical references and index ISBN 0262193663 hc alk paper ISBN 0262691795 pb alk paper 1 Industrial organization Economic Theory 2 Industrial organization Case studies I title HD2326S565 1996 3387dc20 9532647 CIP Fourth printing 1998 Page v For my mother Hadassa Shy and in memory of my father Asher Shy Page vii CONTENTS List of Figures xiii Preface xvii 1 Introduction 1 11 The Study of Industrial Organization 1 12 Law and Economics 5 13 Industrial Organization and International Trade 7 14 References 7 I Theoretical Background 9 2 Basic Concepts in Noncooperative Game Theory 11 21 Normal Form Games 12 22 Extensive Form Games 22 23 Repeated Games 28 24 Appendix Games with Mixed Actions 33 25 Appendix Games with Imperfect Information 37 26 Exercises 40 27 References 42 3 Technology Production Cost and Demand 43 31 Technology and Cost 43 32 The Demand Function 49 33 Appendix Consumer Surplus QuasiLinear Utility 53 34 Exercises 54 Page viii II Market Structures and Organization 57 4 Perfect Competition 63 41 NonIncreasing Returns to Scale 64 42 Increasing Returns to Scale 66 43 MarginalCost Pricing and Social Welfare 68 44 Exercises 69 45 References 70 5 The Monopoly 71 51 The Monopolys ProfitMaximization Problem 72 52 Monopoly and Social Welfare 73 53 Discriminating Monopoly 75 54 The Cartel and the Multiplant Monopoly 78 55 DurableGoods Monopolies 80 56 Appendix The Legal Approach 89 57 Exercises 92 58 References 94 6 Markets for Homogeneous Products 97 61 Cournot Market Structure 98 62 Sequential Moves 104 63 Bertrand Market Structure 107 64 Cournot versus Bertrand 112 65 SerfEnforcing Collusion 115 66 International Trade in Homogeneous Products 120 67 Appendix Cournot with Heterogeneous Firms 126 68 Exercises 128 69 References 131 7 Markets for Differentiated Products 133 71 Two Differentiated Products 135 72 Monopolistic Competition in Differentiated Products 143 73 Location Models 149 74 Appendix Inverting Demand Systems 162 75 Appendix Equilibrium in the Linear City 163 76 Exercises 164 77 References 166 Page ix 8 Concentration Mergers and Entry Barriers 169 81 Concentration Measures 171 82 Mergers 173 83 Entry Barriers 182 84 Entry Deterrence 186 85 Contestable Markets 206 86 Appendix Merger and Antitrust Law 209 87 Appendix Entry Deterrence and Antitrust Law 212 88 Exercises 213 89 References 214 III Technology and Market Structure 219 9 Research and Development 221 91 Classifications of Process Innovation 222 92 Innovation Race 224 93 Cooperation in RD 229 94 Patents 233 95 Licensing an Innovation 239 96 Governments and International RD Races 241 97 Appendix Patent Law 244 98 Appendix Legal Approach to RD Joint Ventures 247 99 Mathematical Appendix 248 910Exercises 248 911 References 250 10 The Economics of Compatibility and Standards 253 101 The Network Externalities Approach 256 102 The Supporting Services Approach 263 103 The Components Approach 269 104 Exercises 276 105 References 276 Page x IV Marketing 279 11 Advertising 281 111 Persuasive Advertising 283 112 Informative Advertising 287 113 Targeted Advertising 290 114 Comparison Advertising 294 115 Other Issues Concerning Advertising 297 116 Appendix Advertising Regulations 300 117 Exercises 302 118 References 304 12 Quality Durability and Warranties 307 121 Personal Income and Quality Purchase 308 122 Quality as Vertical Product Differentiation 310 123 Market Structure Quality and Durability 315 124 The InnovationDurability Tradeoff 317 125 The Market for Lemons 322 126 QualitySignaling Games 327 127 Warranties 330 128 Appendix The Legal Approach to Products Liability 335 129 Exercises 337 1210 References 338 13 Pricing Tactics TwoPart Tariff and PeakLoad Pricing 341 131 TwoPart Tariff 342 132 Nonuniform Pricing 346 133PeakLoad Pricing 348 134 Can Firms Control the Seasons 352 135 Exercises 358 136 References 358 14 Marketing Tactics Bundling Upgrading and Dealerships 361 141 Bundling and Tying 362 142 Killing Off Markets for Used Textbooks 376 143 Dealerships 380 144 Appendix The Legal Approach to Tying 388 145 Appendix Legal Approach to Vertical Restraints 389 146 Exercises 390 147 References 392 Page xi V The Role of Information 393 15 Management Compensation and Regulation 395 151 The PrincipalAgent Problem 396 152 Production with Teams 404 153 Competition and Managerial Compensation 407 154 Why Executives Are Paid More than Workers 413 155 Regulating a Firm under Unknown Cost 416 156 Exercises 419 157 References 420 16 Price Dispersion and Search Theory 421 161 Price Dispersion 421 162 Search Theory 426 163 Mathematical Appendix 432 164 Exercises 432 165 References 433 VI Selected Industries 435 17 Miscellaneous Industries 437 171 Restaurant Economics 438 172 The Airline Industry 440 173 The Fishing Industry 448 174 Public Roads and Congestion 452 175 Exercises 456 176 References 457 Index 459 Page xiii FIGURES 21 The pilot and the terrorist 23 22 Two proper subgames 26 23 Bestresponse functions for the mixedaction extended game 37 24 A game with imperfect information Information sets 38 25 Game with imperfect information Subgames 39 26 Battle of the Sexes in extensive form 42 31 Total average and marginal cost functions 46 32 Duality between the production and cost functions 48 33 Inverse linear demand 49 34 Inverse constantelasticity demand 50 35 Consumers surplus 52 36 Demand generated from a quasilinear utility 54 II1 Commonly assumed and used market structures 61 41 Competitive equilibrium under constant returns to scale 65 42 Decreasing average cost technology 67 43 Marginalcost pricing and social welfare 69 51 The monopolys profit maximizing output 73 52 Monopoly and social welfare 74 53 Monopoly discriminating between two markets 77 54 Durablegood monopoly the case of downward sloping demand 82 55 Durablegood monopoly the case of discrete demand 86 56 Twoperiod game of a durablegood monopoly facing discrete demand 87 Page xiv 61 Cournot bestresponse functions 100 62 Edgeworth Cycles Bertrand competition under capacity constraints 112 63 Residual demand when firms have fixed inventories 114 64 ILs import level under a uniform tariff 123 65 ILs import under the FTA 124 66 The welfare effects of the freetrade agreement 125 71 Approaches to modeling differentiatedproducts industries 134 72 Measuring the degree of product differentiation 137 73 Bestresponse functions for quantity competition in differentiated products 138 74 Bestresponse functions for price competition in differentiated products 139 75 CES indifference curves for N 2 144 76 Decreasing averagecost technology 146 77 Hotellings linear city with two firms 150 78 The position of firms on the unit circle 155 79 Sequentiallocation game 157 710 Discretelocation model 159 711 Undercutproof equilibrium for the discretelocation model 162 712 Existence of equilibrium in the linear city 163 81 Upstream factor suppliers and downstream producers 177 82 Sunk costs and entry barriers 184 83 Sunkcost entry barriers with partial cost recovery 185 84 Incumbents profit levels and capacity choices for different levels of entry cost 191 85 Capacity accumulation and marginal cost 193 86 Relaxing the BainSylos postulate 193 87 Bestresponse functions with fixed capacity 194 88 Capital replacement and entry deterrence 195 89 Judo economics How an entrant secures entry accommodation 199 810 Twoperiod signaling entrydeterrence game 204 811 Contestablemarkets equilibrium 208 91 Classification of process innovation 223 92 RD race between two firms 226 93 Gains and losses due to patent protection 235 Page xv 101 Deriving the demand for telecommunication services 258 102 The PTT profit function in the presence of network externalities 259 103 Twostandard incompatibility equilibrium 261 104 Consumers distribution of tastes 264 105 Equilibrium variety of brandspecific software 267 111 Consumer surplus for a given persuasiveadvertising level 286 112 Equilibrium number of firms placing ads 289 113 Targeted advertising Experienced versus inexperienced consumers 291 114 Informative versus persuasive advertising 294 115 Advertisinginduced demand increase and falling prices 300 121 Horizontal versus vertical differentiation 311 122 Vertical differentiation in a modified Hotelling model 312 123 Determination of the indifferent consumer among brands vertically differentiated on the basis of quality 313 124 Innovation and durability 319 125 The market for lemons Bad cars drive out the good cars 326 131 Quasilinear utility indifference curves 343 132 Pure twopart tariff club charges 345 133 Nonuniform pricing and price discrimination 346 134 Nonuniform price schedule 347 135 Seasonal demand structure and monopoly peakload pricing 349 136 Cost structure of a monopoly selling services in two periods 355 137 Revenue functions for the vertical and horizontal differentiation cases 356 141 Bundling monopoly 362 142 Territorial dealerships in the linear city 385 151 Optimal contract under asymmetric information 403 152 Managers bestresponse function 410 161 Consumers with variable search cost searching for the lowest price 423 162 The determination of the discount and expensive prices 425 163 Prices in a consumersearch model 427 164 Reservationprice strategy 430 Page xvi 171 The equilibrium restaurant price 439 172 Fully connected FC and hubandspoke HS networks 442 173 Evaluation of airfare regulation 447 174 Equilibrium versus optimal highway congestion 455 Page xvii PREFACE If we knew what it was we were doing it would not be called research would it A Einstein Motivation for Writing This Book The motivation for writing this book grew from several years of teaching undergraduate and graduate industrial organization and international trade courses at SUNYAlbany Tel Aviv University and the University of Michigan I felt that for both important fields in economics no theoretical book targeted advanced undergraduate and beginning graduate students Therefore I was guided by my belief that there should not be any necessary correlation between mathematical complexity and theoretical precision That is the purpose of this book is to bring to the advanced student the basic and the latest developments in industrial organization in a very precise manner but without resorting to advanced mathematical techniques By precise I mean that the various market structures and equilibriaand optimal allocations as well as the rules by which firms and consumers actually behaveare always carefully defined I feel that a student of a theoretical course should be able to make precise definitions of what agents actually do and that teaching the student how to precisely define the environment and market structures has nothing to do with getting more mathematical training That is I have attempted to precisely define the equilibria and the models despite the fact that the models are solved for specific examples with no mathematical generality The Level and Prerequisites My intention is to make this book readable to undergraduates who have some training in microeconomics using calculus However in some in Page xviii stances this course can be taught without using calculus see the list of topics in the next section Before reading this book the student should have some experience in maximization techniques for one and twovariables optimization problems Occasionally the student will have to have a very basic knowledge of what probability is and how to calculate the joint probability of two events Nothing in this book requires methods more advanced than the ones I have described Students who did not have any training in microeconomics using calculus may not be able to handle several of the market structures The reader questioning whether this book fits his or her level is advised to look at chapter 3 which reviews the basic microeconomics needed for a comprehensive study of industrial organization Industrial Organization without Calculus Writers of good textbooks should attempt to base most of their arguments on simple logic rather than on long or short derivatives In that respect I admit that I failed to provide the reader with a completely free of calculus book for a very simple reason most of our research and publications are based on calculus and each time I attempted to avoid using calculus I had to reproduce the theory instead of using an existing one The following however is a list of topics that are analyzed without the use of calculus Basic Concepts in Game Theory Chapter 2 Durable Goods Monopolies Subsection 552 Perfect Competition Chapter 4 SelfEnforcing Collusion Section 65 Bertrand Price Competition Section 63 Preferential Trade Agreements among Countries Subsection 662 Sequential Entry to the Linear City Subsection 733 Calculusfree Location Model Subsection 734 Concentration Measures Section 81 Entry Barriers Section 83 Investment in Capital Replacement Subsection 843 Page xix Credible Spatial Preemption Subsection 845 Limit Pricing as Entry Deterrence Subsection 846 Process Innovation Section 91 Innovation Race Section 92 Licensing an Innovation Section 95 International Subsidies for New Product Development Subsection 961 The Economics of Compatibility and Standards Chapter 10 excluding subsection 1011 Advertising Chapter 11 excluding section 111 Quality Durability and Warranties Chapter 12 excluding section 122 Pricing Tactics Chapter 13 excluding section 134 Bundling and Tying Section 141 excluding subsection 1416 Market Segmentation Subsection 1415 Killing Off Used Textbook Markets Section 142 Territorial Dealerships Subsection 1433 The PrincipalAgent Problem Section 151 Regulating a Firm under Unknown Cost Section 155 Why Executives Are Paid More than Workers Section 154 Search Theory Section 162 Restaurant Economics Section 171 Multiproduct Firms Subsection 1721 Price Regulation Subsection 1723 Law and Economics Appendixes Most chapters conclude with nontechnical appendices discussing the major legal issues and laws concerning the topics analyzed in the body of the chapter Page xx To the Instructor Since this book grew out of lecture notes written for upperdivision undergraduate and graduate courses the instructor will I hope find this book convenient to use since almost all derivations are done in the book itself If you are constrained to instruct a course without using calculus then you can teach the list of topics given earlier If you can use some calculus then the amount of material that you can cover depends on your preferences and the length of the course All the theoretical background the student needs for a comprehensive study of this book is provided in the first part In fact not all the material covered in this part is needed to study this book but it is brought up here for the sake of completeness or for those readers who have either an insufficient background in economics or none at all Therefore the instructor is urged to decide on how much time to devote to this preparation part only after having completed the entire plan for this course This theoretical preparation is composed of two chapters Chapter 2 provides all the necessary game theoretic tools needed for the study of this book and for understanding the literature on industrial organization Background in game theory is not needed for reading this chapter and no previous knowledge is assumed The main sections of chapter 2 must be taught before the instructor proceeds with the study of industrial organization Chapter 3 provides most of the basic microeconomics background needed for the study of industrial organization The material covered in this chapter is studied in most intermediate microeconomics and in some managerial economics courses and can therefore be skipped Twosemester course A twosemester course can be logically divided into a more technically marketstructureoriented semester and an applicationoriented semester Thus the first semester should start with game theory chapter 2 continued by the sequence of three chapters dealing with market structure perfect competition chapter 4 monopoly chapter 5 homogeneous products chapter 6 and differentiated products chapter 7 If time is left the first semester may include mergers and entry chapter 8 and research and development chapter 9 For the second semester the instructor is free to select from a wide variety of mostly logically independent topics A possible starting point could be the theory of network economics and standardization chapter 10 continuing with selected topics from the remaining chapters Page xxi advertising chapter 11 durability and quality chapter 12 pricing tactics chapter 13 marketing tactics chapter 14 management and information chapter 15 price dispersion and search theory chapter 16 and the special industries chapter 17 Onesemester course A common mistake at least my mistake in planning a onesemester course would be to treat it as the first semester of a twosemester course When this happens the student is left with the wrong impression that industrial organization deals only with the technical formulation of market structures yet without the knowledge that industrial organization has a lot to say about product design marketing techniques and channels chapters 11 12 13 14 15 and 17 These chapters have many less technically oriented sections with direct applications Some sections rely on the knowledge of Cournot Bertrand and sometime Hotellings market structures and for this reason in a onesemester course I advise the instructor to carefully plan the logical path for this course Finally the material on search theory chapter 16 can be covered with no difficulty Let me summarize then the twosemester course fits the structure and the depth of the coverage of this book The instructor of a onesemester course using this book should study the list of topics covered in the later chapters and then working backwards should determine what is the minimal knowledge of market structures that students need to acquire in order to be able to understand the later chapters New Material Almost by definition a textbook is not intended for presenting newly developed material and ongoing research However during the course of simplifying I was forced to modify or to develop some new concepts For example I felt that it is important to include a location model without using calculus for those courses that do not require the use of calculus However as the reader will find a NashBertrand equilibrium for the discrete location model simply does not exist For this reason I was forced to develop the undercutproof equilibrium concept described in subsection 734 on page 158 Three other topics are also new a the concept of foreclosure developed in subsection 1414 on page 366 b endogenous peakload pricing theory section 134 on page 352 that emphasizes the role of the firm in determining which period would be the peak and which would be the off peak and c targeted and comparison advertising theory sections 113 on page 290 and 114 on page 294 Page xxii Typesetting and Acknowledgments The book was typeset during the months from June 1993 to July 1994 Tel Aviv University and from August 1994 to August 1995 University of Michigan The reader will notice that this book does not have any footnotes Writing a book with no footnotes imposes a significant constraint on the writer because footnotes enable the integration of quasirelated topics into a text However I felt that footnotes impose a great inconvenience to the reader because they tend to disturb the natural flow of reading For this reason I decided to eliminate them As boring as it may sound the following cliché is the whole truth and nothing but the truth Without the help of the people listed below I would not have been able to complete writing this book Therefore I thank Igal Hendel Princeton who was the first person to read the very first draft of several chapters Val Lambson Brigham Young who was the first to test this manuscript in an undergraduate industrial organization class at BYU and was the first to report a success with teaching this material to undergraduates in the United States Tomer Bloomkin a doctoral student at Tel Aviv for reading the manuscript several times and providing many comments and many suggestions throughout that year Henrik Horn Stockholm University for a great many comments and suggestions and for testing the manuscript in a short undergraduate course at Stockholm University Sougata Poddar a doctoral student at CORE Stephen Salant Michigan for a great many comments and illuminating discussions Yossi Spiegel Tel Aviv five anonymous reviewers for The MIT Press and my undergraduate industrial organization and international trade students at Tel Aviv and Michigan I thank Mike Meurer SUNYBuffalo Christopher Proulx Michigan Ennio Stacchetti Michigan and Abi Schwartz Tel Aviv for providing me with comments on selected topics Needless to say I am the only one responsible for all the remaining errors I also would like to thank Martin Osborne McMaster and Hal Varian Berkeley for their most helpful advice and Tianlai Shy for all her help During the preparation of the manuscript I was very fortunate in working with Ann Sochi of The MIT Press to whom I owe many thanks for managing the project in the most efficient way Finally I thank the entire MIT Press team for a fast production of this book ANN ARBOR MICHIGAN AUGUST 1995 ozshyecontauacil Page 1 Chapter 1 Introduction The purpose of an economic theory is to analyze explain predict and evaluate Gathered from Joe Bain Industrial Organization 11 The Study of Industrial Organization 111 Major observations Our approach to analyzing industry behavior is based on four stylized facts Concentration Many industries are composed of few firms Product characteristics Firms in some industries produce homogeneous or almost identical products whereas firms in others distinguish themselves from the competing firms by selling differentiated brands Costly activities Firms in an industry are engaged in repeated costly activities targeted for the purpose of enhancing the sales of their brands In some industries these activities constitute the major cost of the firm and may exceed the cost of producing the product itself These costly activities may include advertising quality control product differentiation costs marketing and dealership costs Research and development Firms allocate resources for inventing cost reducing production technologies as well as new products These resource allocations also include large investments in imitations of technologies invented by rival firms reverse engineering Page 2 It is often thought that these four observations are interrelated Most of the earlier empirical studies in industrial organization focused on running regressions of variables such as profit margins firms size advertising expenditure and research and development RD expenditure on concentration see Goldschmid Mann and Weston 1974 for a summary of these works The purpose of this book is to provide a theoretical linkage of the factors that affect concentration and how concentration affects the strategic behavior of firms The reason why we think of concentration as a major issue of industrial organization theory follows from the failure of the competitive market structure to explain why industries are composed of a few large firms instead of many small firms Thus the theory of competitive market structure although easy to solve for if an equilibrium exists in most cases cannot explain the composition and behavior of firms in the industry Given the noncompetitive behavior of firms markets are also influenced by buyers reactions to firms attempts to maximize profits In this respect our analysis here will have to fully characterize how consumers determine which brands to buy how much to buy and how to search and select the lowest priced brand that fits their specific preferences For this reason the approach we take is mostly a strategic one meaning that both firms and consumers learn the market structure and choose an action that maximizes profit for the firms and utility for the consumers In addition given the complexity of decisions made by strategic noncompetitive firms the issue of the internal organization of firms becomes an important factor affecting their behavior Thus we briefly address the issue of how management structure under conditions of imperfect information affects the performance of the firm in the market Finally we extensively analyze the role of the regulator First from a theoretical point of view we ask whether intervention can increase social welfare under various market structures and firms activities Second we describe and analyze the legal system affecting our industries 112 Schools of thought and methodology The standard approach to the study of industrial organization as laid out by Joe Bain decomposes a market into structure conduct and performance of the market Structure means how sellers interact with other sellers with buyers and with potential entrants Market structure also defines the Product in terms of the potential number of variants in which the product can be produced Market conduct refers to the behavior of the firms in a given market structure that is how firms determine their price policy sales and promotion Finally performance refers to the Page 3 welfare aspect of the market interaction That is to determine performance we measure whether the interaction in the market leads to a desired outcome or whether a failure occurs that requires the intervention of the regulator Many aspects of performance are discussed in this book First is the technology efficient in the sense of whether it is operated on an optimal costminimizing scale Second does the industry produce a socially optimal number of brands corresponding to consumers preferences and the heterogeneity of the consumers Third are the firms dynamically efficientdo they invest a proper amount of resources in developing new technologies for current and future generations All these efficiency requirements are generally summarized by a particular social welfare function that can combine the tradeoff among the different efficiency criteria For example the welfare of consumers who have preferences for variety increases with the number of brands produced in an industry However if each brand is produced by a different factory where each factory is constructed with a high fixedcost investment then it is clear that from a technical point of view the number of brands produced in an industry should be restricted Hence there will always be a tradeoff between technical efficiency and consumer welfare that will require defining a welfare function to determine the optimal balance between consumer welfare and efficient production patterns In 1939 Edward Mason published a very influential article emphasizing the importance of understanding the marketspecific causes of noncompetitive behavior In that article Mason discussed the methodology for studying the various markets It goes without saying that a realistic treatment of these questions necessitates the use of analytical tools which are amenable to empirical application The problem as I see it is to reduce the voluminous data concerning industrial organization to some sort of order through a classification of market structures Differences in market structure are ultimately explicable in terms of technological factors The economic problem however is to explain through an examination of the structure of markets and the organization of firms differences in competitive practices including price production and investment policies Thus Mason argued that to be able to understand different degrees of competition in different markets the researcher would have to analyze the different markets using different assumed market structures The reader will appreciate this methodology after reading this book where we try to fit an appropriate market structure to the studied specific Page 4 market where the variety of market structures are defined and developed in part II In his article Mason emphasized the importance of understanding sources of market power market control in his language in order to understand how prices are determined in these markets price policy in his language A firm may have a price policy by reason of the existence of rivals of whose action it must take account of the desirability of considering the effect of present upon future price of the possibility of competing in other ways than by price and for many other reasons Mason continues and hints at how the degree of industry concentration is correlated with noncompetitive behavior The size of a firm influences its competitive policies in a number of waysThe scale of its purchases and sales relative to the total volume of transactionsthe absolute size of a firm as measured by assets employees or volume of salesare also relevant to price and production policiesSelling practices at the disposal of the large firm may be beyond the reach of its smaller competitorsThe size of a firm likewise influences its reaction to given market situations Analysts of industrial organization after Mason continued mostly to use a descriptive language but later ones used price theory sometimes referred to as the Chicago School The Chicago price theory approach conceded that monopoly is possible but contended that its presence is much more often alleged than confirmed When alleged monopolies are genuine they are usually transitory with freedom of entry working to eliminate their influence on price and quantities within a fairly short time period see Reder 1982 Thus the socalled Chicago School was not very supportive of the persistentmarketpower approach that constituted Bains major theory of entry barriers The fast development of game theory in the 1970s gave a push to the strategic approach to industrial organization and later to strategic international trade analysis Unlike the competitive markets approach the strategic approach models the firms on the assumption that they and other firms can affect the market outcome consisting of prices quantities and the number of brands In addition game theory provided the tools for analyzing dynamic scenarios such as how established firms react to a threat of entry by potential competitors Our approach does not attempt to represent any particular school of thought In fact the main purpose of this book is to demonstrate Page 5 that there is no general methodology for solving problems hence each observation may have to be worked out in a different model Thus each time we address a new observation we generally construct a special ad hoc model where the term ad hoc should not be given a negative connotation To the contrary the ad hoc modeling methodology frees the researcher from constraining the theory to temporary fashions which are given a priority in the scientific literature and allows the scientist to concentrate on the merit of the model itself where merit means how well the theory or the model explains the specific observation that the scientist seeks to explain Nevertheless the reader will discover that the strategic gametheoretic approach is the dominant one in this book 12 Law and Economics The legal structure governing the monitoring of the industry is called antitrust law The word trust reflects the spirit of the laws aiming at any form of organization trust communication and contract among firms that would impede competition In this book we confine the discussion of the legal aspects of the industry mainly to US law I chose to deal with US law since it is perhaps the most advanced in terms of achieving competition and the restraints of monopoly power Although not the oldest the US antitrust system seems to be the most experienced one in terms of famous court cases that put the legal system into effect For example the Restrictive Trade Practices Act which is the British equivalent of the 1890 Sherman Act regarding cartel prohibition was enacted a very long time after the Sherman Act in 1956 to be precise In other words the US was and remains a leader in antitrust legislation It is interesting to note that in the United States real prices of products tend to be the lowest in the world However the United States also has the most restrictive antitrust regulation structure in the world Hence although it is commonly argued that market intervention in the form of regulation results in higher consumer prices here we observe that antitrust regulation is probably the cause for low consumer prices in the United States For this reason the study of the US antitrust systems is an integral part of the study of industrial organization especially for those students from countries with less competitive markets Several chapters in this book conclude with appendixes discussing the legal matters related to the topics analyzed in the theoretical part of the chapter In these appendixes references are always made to the law itself and to its historical origin Court cases are not discussed in this book since they are analyzed in a large number of lawandeconomics textbooks for example Asch 1983 Gellhorn 1986 and Posner 1977 Page 6 121 The development of the antitrust legal system It is not surprising that when the Sherman Antitrust Act was passed in 1890 economists were almost unanimously opposed to it on the basis that trust busting would involve a loss of the efficiency advantages of combinations or trusts West 1987 Interestingly after a decade of strict enforcement of the older mergers guidelines issued by the Federal Trade Commission the newer 1984 guidelines have brought back the efficiency argument as an argument for merger in medium concentrated industries The reader interested in learning the development of the antitrust laws should not miss reading Bork 1978 According to Bork the major development and the entire set of disputes and theoretical conjectures were all formed during the period from 1890 Sherman Act to 1914 Clayton Act and the Federal Trade Commission Act The Sherman Act of 1890 was intended to strike at cartels horizontal mergers of monopolistic nature and predatory business activities Section 1 of this act stated that Every contract combination in the form of trust or otherwisein restraint of trade or commerceis hereby declared to be illegal Earlier court interpretations followed section 1 of the act precisely as stated but soon began to adhere to the rule of reason in which not every act of merger was considered as a restraint of trade The courts began identifying which restraints were reasonable and which were not In 1911 a major ruling based on the Sherman Act was handed down wherein some of Standard Oils activities were found to be illegal leading to the dissolution of this giant into thirty companies In that period American Tobacco also broke up A largescale dissolution occurred again in 1982 when ATT responded to pressure to break up into the seven baby Bell companies and ATT The ATT breakup was effected by consent decree and not by litigation The search for which restraints of trade are reasonable led to a more refined legislation the Clayton Act of 1914 in which price discrimination exclusive dealing and corporate stock acquisition that may lead to reduced competition were declared illegal The Federal Trade Commission Act of 1914 mandated the FTC to categorize and identify what constitute unfair methods of competition 122 The Per Se versus the Rule of Reason approaches In all the lawandeconomies appendixes we make a use of two methods of court ruling in antitrust cases the per se rule and the rule of reason Bork 1978 defines the rule of reason as a set of general categories that are given content by ideas about the proper goals of the law economics and the requirement of the judicial process In other words court rulings consist of two major categories a business behavior that Page 7 is illegal per se and b business behavior that is judged by standards of the partys intent or the effect the behavior is likely to have For our purposes we will refer to the rule of reason as category b Bork 1978 regards the per se rule as containing a degree of arbitrariness The per se rule implies that the judgment is handed down on the basis of the inherent effect of the act committed by the accused party That is to have a particular behavior declared illegal per se the plaintiff needs only to prove that it occurred The per se rule is justified in cases where the gains associated from the imposition of the rule will far outweigh the losses since significant administrative costs can be saved That is the advantage of the per se rule is that the particular case need not be identified since the act itself is assumed to be illegal 13 Industrial Organization and International Trade In this book the reader will find a wide variety of international issues for the simple reason that international markets should not be very different from national markets Thus one might expect that concentration would characterize international markets as well as national markets As a result of this rather late recognition that international trade can be characterized by oligopolistic market structures a tremendous amount of literature emerged during the 1980s see Krugman 1989 Once this newer trade theory picked up a broad new array of issues had to be analyzed The first was how can international trade in differentiated products be explained by a monopolistic competition market structure Then what are the implications of oligopolistic international market structures for the gains from the imposition of trade barriers Whereas earlier writers got excited by learning that countries have a lot to gain when imposing trade restrictions or allowing subsidization of industries competing in internationally oligopolistic markets later writers have managed to calm down this new wave of protectionism by demonstrating that any trade policy recommended under a particular market structure may not be recommended under a different market structure Thus since it is hard to estimate what the ongoing market structure is and the form of competition of a particular market it may be better that governments refrain from intervention at all These later papers have somewhat mitigated the strong policy actions recommended by the early strategic trade literature 14 References Asch P 1983 Industrial Organization and Antitrust Policy New York John Wiley Sons Page 8 Bain J 1968 Industrial Organization 2nd ed New York John Wiley Sons Bork R 1978 The Antitrust Paradox New York Basic Books Gellhorn E 1986 Antitrust Law and Economics in a Nutshell St Paul Minn West Publishing Goldschmid H H Mann and J Weston 1974 Industrial Concentration The New Learning Boston Little Brown Krugman P 1989 Industrial Organization and International Trade In Handbook of Industrial Organization edited by R Schmalensee and R Willig Amsterdam NorthHolland Mason E 1939 Price and Production Policies of LargeScale Enterprise American Economic Review 29 pt 2 6174 Posner R 1977 Economic Analysis of Law Boston Little Brown Reder M 1982 Chicago Economics Performance and Change Journal of Economic Literature 20 138 West E 1987 Monopoly In The New Palgrave Dictionary of Economics edited by J Eatwell M Milgate and P Newman New York The Stockton Press Page 9 PART I THEORETICAL BACKGROUND GAME THEORY AND MICROECONOMICS Page 11 Chapter 2 Basic Concepts in Noncooperative Game Theory If you know the enemy and know yourself you need not fear the result of a hundred battles If you know yourself but not the enemy for every victory gained you will also suffer a defeat If you know neither the enemy nor yourself you will succumb in every battle All men can see these tactics whereby I conquer but what none can see is the strategy out of which victory is evolved Sun Tzu The Art of War 490 BC Game theory sometimes referred to as Interactive Decision Theory is a collection of tools for predicting outcomes for a group of interacting agents where an action of a single agent directly affects the payoffs welfare or profits of other participating agents The term game theory stems from the resemblance these tools to sports games eg football soccer pingpong and tennis as well as to social games eg chess cards checkers and Diplomacy Game theory is especially useful when the number of interactive agents is small in which case the action of each agent may have a significant effect on the payoff of other players For this reason the bag of tools and the reasoning supplied by game theory have been applied to a wide variety of fields including economics political science animal behavior military studies psychology and many more The goal of a gametheoretic model is to predict the outcomes a list of actions Page 12 adopted by each participant given the assumed incentives of the participating agents Thus game theory is extremely helpful in analyzing industries consisting of a small number of competing firms since any action of each firm whether price choice quantity produced research and development or marketing techniques has strong effects on the profit levels of the competing firms As the title of this chapter suggests our analyses focus only on noncooperative games We generally distinguish between two types of game representations normal form games analyzed in section 21 and extensive form games analyzed in section 22 Roughly speaking we can say that in normal form games all players choose all their actions simultaneously whereas in extensive form games agents may choose their actions in different time periods In addition we distinguish between two types of actions that players can take a pure action where a player plays a single action from the players set of available actions and a mixed action where a player assigns a probability for playing each action say by flipping a coin Our entire analysis in this book is confined to pure actions However for the sake of completeness mixed actions are analyzed in an appendix section 24 Finally information plays a key role in game theory as well as in real life The most important thing that we assume is that the players that we model are at least as intelligent as economists are That is the players that we model have the same knowledge about the structure the rules and the payoffs of the game as the economist that models the game does Also important our analysis in this chapter is confined to games with perfect information Roughly this means that in perfect information games each player has all the information concerning the actions taken by other players earlier in the game that affect the players decision about which action to choose at a particular time Games under imperfect information are not used in this book however we introduce them in an appendix section 25 for the sake of completeness 21 Normal Form Games Our first encounter with games will be with normal form games In normal form games all the players are assumed to make their moves at the same time 211 What is a game The following definition provides three elements that constitute what we call a game Each time we model an economic environment in a gametheoretic framework we should make sure that the following three Page 13 elements are clearly stipulated Definition 21 A normal form game is described by the following 1 A set of N players whose names are listed in the set 2 Each player i has an action set Ai which is the set of all actions available to player i Let a denote a particular action taken by player i Thus player is action set is a list of all actions available to player i and hence where ki is the number of actions available to player i Let be a list of the actions chosen by each player We call this list of actions chosen by each player i an outcome of the game 3 Each player i has a payoff function πi which assigns a real number πia to every outcome of the game Formally each payoff function πi maps an Ndimensional vector a a1 aN the action of chosen by each player and assigns it a real number πia A few important remarks on the definition of a game follow 1 It is very important to distinguish between an action set Ai which is the set of all actions available to a particular player i and an outcome a which is a list of the particular actions chosen by all the players 2 Part 2 of Definition 21 assumes that the each player has a finite number of actions that is that player i has ki actions in the action set Ai However infinite action sets are commonly used in industrial organization For example often we will assume that firms choose prices from the set of nonnegative real numbers 3 We use the notation list of elements to denote a set where a set eg an action set contains elements in which the order of listing is of no consequence In contrast we use the notation list to denote a vector where the order does matter For example an outcome is a list of actions where the first action on the list is the action chosen by player 1 the second by player 2 and so on 4 The literature uses the term action profile to describe the list of actions chosen by all players which is what we call an outcome For our purposes there is no harm in using the term outcome instead of the term action profile for describing this list of actions However Page 14 if games involve some uncertainty to some players these two terms should be distinguished since under uncertainty an action profile may lead to several outcomes see for example mixed actions games described in the appendix Section 24 5 In the literature one often uses the term stoutly instead of the term action and therefore strategy set instead of action set since in a normal form game there is no distinction between the two terms However when we proceed to analyze extensive form games section 22 the term strategy is given a different meaning than the term action The best way to test whether Definition 21 is clear to the reader is to apply it to a simple example A simple way to describe the data that define a particular game is to display them in a matrix form Consider the following game described in Table 21 We now argue that Table 21 Country 2 WAR PEACE Country 1 WAR 1 1 3 0 PEACE 0 3 2 2 Table 21 PeaceWar game contains all the data needed for properly defining a game according to Definition 21 First we have two players N 2 called country 1 and 2 Second the two players happen to have the same action sets A1 A2 WAR PEACE There are exactly four outcomes for this game WAR WAR WAR PEACE PEACE WAR PEACE PEACE Third the entries of the matrix ie the four squares contain the payoffs to player 1 on the lefthand side and to player 2 on the righthand side corresponding to the relevant outcome of the game For example the outcome a WAR PEACE specifies that player 1 opens a war while player 2 plays peace The payoff to player 1 from this outcome is π1a π1WAR PEACE 3 Similarly the payoff to player 2 is π2a π2WAR PEACE 0 since country 2 does not defend itself The story behind this game is as follows If both countries engage in a war then each country gains a utility of 1 If both countries play PEACE then each country gains a utility of 2 If one country plays WAR while the other plays PEACE then the aggressive country reaches the highest possible utility since it wins a war against the nonviolent country with no effort Under this outcome the utility of the pacifist country should be the lowest equal to zero in our example Page 15 In the literature the game described in Table 21 is commonly referred to as the Prisoners Dilemma game Instead of having two countries fighting a war consider two prisoners suspected of having committed a crime for which the police lack sufficient evidence to convict either suspect The two prisoners are put in two different isolated cells and are offered a lower punishment or a higher payoff if they confess of having jointly committed this crime If we replace WAR with CONFESS and PEACE with NOT CONFESS we obtain the socalled Prisoners Dilemma game In the present analysis we refrain from raising the question whether the game described in Table 21 is observed in reality or not or whether the game is a good description of the world Instead we ask a different set of questions namely given that countries in the world behave like those described in Table 21 can we the economists or political scientists predict whether the world will end up in countries declaring war or declaring peace In order to perform this task we need to define equilibrium concepts 212 Equilibrium concepts Once the game is properly defined we can realize that games may have many outcomes Therefore by simply postulating all the possible outcomes four outcomes in the game described in Table 21 we cannot make any prediction of how the game is going to end For example can you predict how a game like the one described in Table 21 would end up Will there be a war or will peace prevail Note that formulating a game without having the ability to predict implies that the game is of little value to the researcher In order to make predictions we need to develop methods and define algorithms for narrowing down the set of all outcomes to a smaller set that we call equilibrium outcomes We also must specify properties that we find desirable for an equilibrium to fulfill Ideally we would like to find a method that would select only one outcome If this happens we say that the equilibrium is unique However as we show below the equilibrium concepts developed here often fail to be unique Moreover the opposite extreme may occur where a particular equilibrium may not exist at all A game that cannot be solved for equilibria is of less interest to us since no reallife prediction can be made Before we proceed to defining our first equilibrium concept we need to define one additional piece of notation Recall that an outcome of the game a a1 ai aN is a list of what the N players are doing playing Now pick a certain player whom we will call player i eg i can be player 1 or 89 or N or any player Remove from the outcome Page 16 a the action played by player i himself Then we are left with the list of what all players are playing except player i which we denote by Formally Note that after this minor surgical operation is performed we can still express an outcome as a union of what action player i plays and all the other players actions That is an outcome a can be expressed as Equilibrium in dominant actions Our first equilibrium concept called equilibrium in dominant strategies is a highly desirable equilibrium in the sense that if it exists it describes the most intuitively plausible prediction of what players would actually do The following definition applies for a single player in the sense that it classifies actions in a players action set according to a certain criterion Definition 22 A particular action is said to be a dominant action for player i if no matter what all other players are playing playing always maximizes player is payoff Formally for every choice of actions by all players except i For example Claim 21 In the game described in Table 21 the action a1 WAR is a dominant action for player 1 Proof It has to be shown that no matter what player 2 does player 1 is always better off by starting a war Thus we have to scan over all the possible actions that can be played by player 2 If player 2 plays a2 WAR then Also if player 2 plays a2 PEACE then Page 17 Similarly since the game is symmetric meaning that renaming player 1 as player 2 and vice versa does not change players payoffs the reader can establish that a2 WAR is a dominant action for player 2 We now turn to defining our first equilibrium concept An equilibrium in dominant actions is simply an outcome where each player plays a dominant action Formally Definition 23 An outcome where for every i 1 2 N is said to be an equilibrium in dominant actions if is a dominant action for each player i Clearly since WAR is a dominant action for each player in the game described in Table 21 the outcome a1 a2 WAR WAR is an equilibrium in dominant actions Although an equilibrium in dominant actions constitutes a very reasonable prediction of how players may interact in the real world unfortunately this equilibrium does not exist for most games of interest to us To demonstrate this point let us analyze the following Battle of the Sexes game described in Table 22 The intuition behind this rather Rachel OPERA ω FOOTBALL φ Jacob OPERAω 2 1 0 0 FOOTBALL φ 0 0 1 2 Table 22 Battle of the Sexes romantic Battle of the Sexes game is that it is relatively important for Jacob and Rachel to be together That is assuming that the payoffs to the players in Table 22 represent utilities to each player under each outcome each player gains the lowest possible utility when the player goes alone to one of these entertainment events Both of them gain a higher utility if they go together to one of these events However comparing the two outcomes where the players are together we can observe that Jacob prefers the OPERA whereas Rachel prefers FOOTBALL Thus the Battle of the Sexes is sometimes referred to as a coordination game The Battle of the Sexes game exhibited in Table 22 describes some reallife situations For example in chapter 10 we analyze economies in which products operate on different standards such as different TV systems The Battle of the Sexes game happens to be an ideal theoretical framework to model two firms with two available actions choose standard 1 or standard 2 Failure to have both firms choosing the same standard may result in having consumers reject the product thereby leaving the two firms with zero profits Page 18 After formulating the Battle of the Sexes game we now seek to find some predictions for this game However the reader will probably be disappointed to find out that Claim 22 There does not exist an equilibrium in dominant actions for the Battle of the Sexes game Proof It is sufficient to show that one of the players does not have a dominant action In this case there cannot be an equilibrium in dominant actions since one player will not have a dominant action to play Therefore it is sufficient to look at Jacob If Rachel chooses aR ω then Jacob would choose ω because However when Rachel goes to a football game aR φ then Jacob would choose φ because So we have shown that one player does not have a dominant action and this suffices to conclude that Definition 23 cannot be applied hence there does not exist an equilibrium in dominant actions for the Battle of the Sexes game Nash equilibrium NE So far we have failed to develop an equilibrium concept that would select an outcome that would be a reasonable prediction for this model In 1951 John Nash provided an existence proof for an equilibrium concept earlier used by Cournot when studying duopoly that has become the most commonly used equilibrium concept in analyzing games Definition 24 An outcome where for every i 1 2 N is said to be a Nash equilibrium NE if no player would find it beneficial to deviate provided that all other players do not deviate from their strategies played at the Nash outcome Formally for every player i i 1 2 N The general methodology for searching which outcomes constitute a NE is to check whether players benefit from a unilateral deviation from a certain outcome That is to rule out an outcome as a NE we need only Page 19 demonstrate that one of the players can increase the payoff by deviating to a different action than the one played in this specific outcome assuming that all other players do not deviate Once we find an outcome in which no player can benefit from any deviation from the action played in that outcome we can assert that we found a NE outcome We continue our discussion of the NE with the investigation of the relationship between Nash equilibrium and equilibrium in dominant actions To demonstrate the relationship between the two equilibrium concepts we first search for the NE outcomes for the game described in Table 21 Recall that we have already found that WAR WAR is an equilibrium in dominant actions but can this fact help us in searching for a NE for this game Not surprisingly yes it can Since an equilibrium in dominant actions means that each player plays a dominant action no player would find it beneficial to deviate no matter how the others play In particular no player would deviate if the other players stick to their dominant actions Hence Proposition 21 An equilibrium in dominant actions outcome is also a NE However a NE outcome need not be an equilibrium in dominant actions Altogether we have it that WAR WAR is a NE for the game described in Table 21 We leave it to the reader to verify that no other outcome in this game is a NE Therefore this equilibrium is called unique The second part of Proposition 21 follows from the Battle of the Sexes game where there exist two NE but there does not exist an equilibrium in dominant actions Multiple Nash equilibria We now demonstrate that a Nash equilibrium need not be unique For example applying Definition 24 to the Battle of the Sexes game yields Claim 23 The Battle of the Sexes game described in Table 22 has two Nash equilibrium outcomes OPERA OPERA and FOOTBALL FOOTBALL Proof To prove that ω ω is a NE we have to show that no player would benefit from deviation given that the other does not deviate In this game with two players we have to show that given that aR ω player J would play aJ ω and that given that aJ ω player R would play aR ω These two conditions follow from Page 20 Using the same procedure it can be easily shown that the outcome φ φ is also a NE Finally we need to show that the other two outcomes ω φand φ ω are not NE However this follows immediately from 21 Nonexistence of a Nash equilibrium So far we have seen examples where there is one or more NE That is as in the Battle of the Sexes game displayed in Table 22 it is always possible to find games with multiple NE If the equilibrium is not unique the model has a low prediction power In contrast Table 23 demonstrates a game where a Nash equilibrium does not exist Therefore consider the variant of the Battle of the Sexes game after thirty years of marriage The intuition behind the game described in Table 23 is that after Rachel OPERA ω FOOTBALL φ Jacob OPERA ω 2 0 0 2 FOOTBALL φ 0 1 1 0 Table 23 Nonexistence of a NE in pure actions thirty years of marriage Rachels desire for being entertained together with Jacob has faded however Jacobs romantic attitude remained as before and he would always gain a higher utility from being together with Rachel rather than alone Proposition 22 The game described in Table 23 does not have a NE Proof We must prove that each outcome is not a NE That is in each of the four outcomes at least one of the player would find it beneficial to deviate 1 For the ω ω outcome Hence Rachel would deviate toaR φ 2 For the φ ω outcome Hence Jacob would deviate to aJ ω 3 For the φ φ outcome Hence Rachel would deviate to aR ω 4 For the ω φ outcome Hence Jacob would deviate toaJ φ Using bestresponse functions to solve for NE We now develop a tool called bestresponse functions that facilitates the search for NE Page 21 Definition 25 1 In a twoplayer game the bestresponse function of player i is the function Riaj that for every given action aj of player j assigns an action ai Riaj that maximizes player is payoff πiai aj 2 More generally in an Nplayer game the bestresponse function of player i is the function that for given actions of players 1 2 i 1 i 1 N assigns an action that maximizes player is payoff Let us now construct the bestresponse functions for Jacob and Rachel described in the Battle of the Sexes game given in Table 22 It is straightforward to conclude that That is if Rachel plays ω Jacobs best response is to play ω and if Rachel plays φ Jacobs best response is to play φ and so on Now the importance of learning how to construct bestresponse functions becomes clear in the following proposition Proposition 23 If is a Nash equilibrium outcome then for every player i Proof By Definition 24 in a NE outcome each player does not benefit from deviating from the strategy played in a NE outcome given that all other players do not deviate Hence by Definition 25 each player is on her bestresponse function That is in a NE outcome each player chooses an action that is a best response to the actions chosen by other players in a NE Proposition 23 is extremely useful in solving for NE in a wide variety of games and will be used extensively The procedure for finding a NE is now very simple First we calculate the bestresponse function of each player Second we check which outcomes lie on the bestresponse functions of all players Those outcomes that we find to be on the bestresponse functions of all players constitute the NE outcomes For example in the Battle of the Sexes game 22 implies that outcomes ω ω and φ φ each satisfy both players bestresponse functions and therefore constitute NE outcomes Page 22 213 Welfare comparisons among outcomes So far our analysis has concentrated on defining equilibrium concepts that enable us to select equilibrium outcomes for predicting how players would end up acting when facing similar games in the real world However we have not discussed whether the proposed equilibria yield efficient outcomes That is we wish to define an efficiency concept that would enable us to compare outcomes from a welfare point of view In particular using the Pareto efficiency criterion we wish to investigate whether there are outcomes that yield higher payoff levels to some players without reducing the payoffs of all other players For example in the PeaceWar game of Table 21 the outcome PEACE PEACE yields higher payoffs to both players compared with the outcome WAR WAR In this case we say that the outcome PEACE PEACE Pareto dominates the outcome WAR WAR Formally Definition 26 1 The outcome â Pareto dominates the outcome a also called Pareto superior to a if a For every player i and b there exists at least one player j for whom 2 An outcome a is called Pareto efficient also called Pareto optimal if there does not exist any outcome which Pareto dominates the outcome a 3 Outcomes a and ã are called Pareto noncomparable if for some player i but for some other player j For example in the PeaceWar game the outcomes WAR PEACE and PEACE WAR are Pareto noncomparable In the Battle of the Sexes game of Table 22 the outcomes OPERA FOOTBALL and FOOTBALL OPERA are Pareto dominated by each of the other two outcomes The outcomes OPERA OPERA and FOOTBALL FOOTBALL are Pareto efficient and are also Pareto noncomparable 22 Extensive Form Games Our analysis so far has concentrated on normal form games where the players are restricted to choosing an action at the same time In this section we analyze games in which players can move at different times and more than once Such games are called extensive form games Extensive form games enable us to introduce timing into the model Page 23 Before going to the formal treatment let us consider the following example A terrorist boards a flight from Minneapolis to New York After thirty minutes after reaching a cruising altitude of thirty thousand feet the terrorist approaches the pilot and whispers to the pilot that she will explode a bomb if the pilot does not fly to Cuba Figure 21 describes the PilotTerrorist game One player is the pilot and the other is the Figure 21 The pilot and the terrorist terrorist The game is represented by a tree with a starting decision node point I other decision nodes IIN and IIC and terminal nodes end points Note that in some literature the term vertex vertices is used in place of the term nodes The branches connecting decision nodes and decision nodes to terminal nodes describe actions available to the relevant player on a particular decision node In this PilotTerrorist game after hearing the terrorists threat the pilot gets to be the player to choose an action at the starting node At the starting node the pilots action set is given by Depending on what action is chosen by the pilot the terrorist has her turn to move at node IIC or IIN The terrorists action set is at the node IIC and at the node IIN In this simple game the terrorists action sets happen to be the same at both nodes but this need not always be the case We can now give a formal definition to extensive form games with perfect information Extensive form games with imperfect information are defined in Definition 217 on page 38 Definition 27 An extensive form game is 1 A game tree containing a starting node other decision nodes terminal nodes and branches linking each decision node to successor nodes Page 24 2 A list of players indexed by i i 1 2 N 3 For each decision node the name of the player entitled to choose an action 4 For each player i a specification of is action set at each node that player i is entitled to choose an action 5 A specification of the payoff to each player at each terminal node 221 Defining strategies and outcomes in extensive form games Our preliminary discussion of extensive form games emphasized that a player may be called to choose an action more than once and that each time a player chooses an action the player has to choose an action from the action set available at that particular node Therefore we need to define the following term Definition 28 A strategy for player i denoted by si is a complete plan list of actions one action for each decision node that the player is entitled to choose an action Thus it is important to note that a strategy is not what a player does at a single specific node but is a list of what the player does at every node where the player is entitled to choose an action What are the strategies available to the terrorist in the PilotTerrorist game described in Figure 21 Since the terrorist may end up in either node IIC or IIN a strategy for the terrorist would be a specification of the precise action she will be taking at each node That is although it is clear that the terrorist will reach either node IIC or IIN but not both a strategy for this player must specify what she will do at each of the two nodes Therefore the terrorist has four possible strategies given by B B B NB NB B NB NB where the first component refers to the terrorists action in node IIC and the second component refers to her action at node IIN Since the pilot is restricted to making a move only at node I and since his action set has two possible actions this game has eight outcomes given by NY B B NY B NB NY NB B NY NB NB C B B C B NB C NB B C NB NB 222 A normal form representation for extensive form games Now that the game is well defined we seek to find some predictions The first step would be to search for a Nash equilibrium Recalling our definition of Nash equilibrium Definition 24 in extensive form games Page 25 we look for a Nash equilibrium in strategies where each player cannot increase the payoff by unilaterally deviating from the strategy played at the NE outcome It turns out that in many instances transforming an extensive form game into a normal form makes it easier to find the Nash equilibria Table 24 provides the normal form representation for the Pilot Terrorist game described in Figure 21 Table 24 shows that there are three Nash Terrorist B B B NB NB B NB NB Pilot NY 1 1 2 0 1 1 2 0 CUBA 1 1 1 1 1 1 1 1 Table 24 Normal form representation of the PilotTerrorist game equilibrium outcomes for this game NY NB NB NY B NB and CUBA NB B Note that here as in the Battle of the Sexes game multiple NE greatly reduce our ability to generate predictions from this game For this reason we now turn to defining an equilibrium concept that would narrow down the set of NE outcomes into a smaller set of outcomes In the literature an equilibrium concept that selects a smaller number of NE outcomes is called a refinement of Nash equilibrium which is the subject of the following subsection 223 Subgames and subgame perfect equilibrium In this subsection we define an equilibrium concept that satisfies all the requirement of NE see Definition 24 and has some additional restrictions This equilibrium concept may be helpful in selecting a smaller set of outcomes from the set of NE outcomes by eliminating some undesirable NE outcomes Before we proceed to the formal part let us go back to the PilotTerrorist game and look at the three NE outcomes for this game Comparing the three NE outcomes do you consider any equilibrium outcomes to be unreasonable What would you suggest if the pilot were to hire you as her strategic adviser Well you would probably tell the pilot to fly to New York Why By looking at the terrorists payoffs at the terminal nodes in Figure 21 we can see that if the pilot flies to NEW YORK the terrorist will NOT BOMB a payoff of πt 0 compared with πt 1 if she does and the pilot will gain a payoff of πp 2 compared with a payoff of πp 1 for flying to Cuba In other words after the pilot flies to any destination New York or Cuba the terrorists payoff is maximized by choosing the NOT BOMB action From Page 26 this we conclude that the limitation of the NE concept is that it cannot capture the pilots ability to predict that the terrorist will not have the incentive to explode the bomb once the plane arrives in New York in to Cuba More precisely under the NE outcomes CUBA NB B and NY B NB the terrorist seems to be pulling what game theorists call an incredible threat since the terrorists payoffs at the terminal nodes indicate that once reaching either node IIC or IIN the terrorist will not explode the bomb We now want to formalize an equilibrium concept that would exclude the unreasonable Nash equilibria In particular we look for an equilibrium concept that would exclude outcomes where the terrorist commits herself to the BOMB action since such an action is incredible Moreover we seek to define an equilibrium concept where the player who moves first the pilot in our case would calculate and take into account how subsequent players the terrorist in the present case would respond to the moves of the players who move earlier in the game Hence having computed how subsequent players would respond the first player can optimize by narrowing down the set of actions yielding higher payoffs In the PilotTerrorist example we wish to find an equilibrium concept that would generate a unique outcome where the pilot flies to New York We first define a subgame of the game Definition 29 A subgame is a decision node from the original game along with the decision nodes and terminal nodes directly following this node A subgame is called a proper subgame if it differs from the original game Clearly the PilotTerrorist game has three subgames One is the game itself whereas the other two are proper subgames with nodes IIC and IIN as starting nodes The two proper subgames are illustrated in Figure 22 Figure 22 Two proper subgames In 1965 Rheinhard Selten proposed a refinement of the NE concept defined as follows Page 27 Definition 210 An outcome is said to be a subgame perfect equilibrium SPE if it induces a Nash equilibrium in every subgame of the original game Definition 210 states that a SPE outcome is a list of strategies one for each player consisting of players actions that constitutes a NE at every subgame In particular a SPE outcome must be a NE for the original game since the original game is a subgame of itself Note that in each subgame the action NB is a NE We now seek to apply Definition 210 in order to solve for a SPE of the PilotTerrorist game Claim 24 The outcome NY NB NB constitutes a unique SPE for the PilotTerrorist game Proof Since a SPE is also a NE for the original game it is sufficient to look at the three NE outcomes of the original game given by NY B NB Cuba NB B and NY NB NB Next each proper subgame has only one NE namely the terrorist chooses NB Hence given that a SPE outcome must be a NE for every subgame we conclude that the outcomes NY B NB Cuba NB B are not SPE Finally the outcome NY NB NB is a SPE since it is a NE for the original game and the outcome action NB is a unique NE for every proper subgame Thus we have shown that using the SPE refines the NE in the sense of excluding some outcomes which we may consider unreasonable We conclude this discussion of the SPE by describing the methodologies commonly used for finding SPE outcomes The general methodology for finding the SPE outcomes is to use backward induction meaning that we start searching for NE in the subgames leading to the terminal nodes Then we look for NE for the subgames leading the subgames leading to the terminal nodes taking as given the NE actions to be played in the last subgames before the terminal nodes Then continuing to solve backwards we reach the starting node and look for the action that maximizes player 1s payoff given the NE of all the proper subgames Note that the backward induction methodology is particularly useful when the game tree is long Finally another common methodology is to first find the NE outcomes for the game say by transforming the extensive form representation into a normal form representation see subsection 222 Then once we have the set of all NE outcomes we are left to select those outcomes that are also NE for all subgames This can be done by trial and error or as we do in the proof of Claim 24 by ruling out the NE outcomes of the original game that are not NE for some proper subgames Page 28 23 Repeated Games Repeated games are used only once in this book in section 65 where we analyze collusion among firms under imperfect competition A repeated game is a oneshot game that is identically repeated more than once The importance of analyzing repeated games is that under certain circumstances cooperative outcomes which are not equilibrium outcomes under a oneshot game can emerge as equilibrium outcomes in a repeated noncooperative game A repeated game is a special kind of an extensive form game in which each period all players move simultaneously and each players action set does not vary with time in a more general extensive form game actions sets may differ from one time period to another More precisely a repeated game is a oneshot game see Definition 21 that is repeated for several periods where the repeated game is played once in each period Each period after the game is played the players move to the next period In a subsequent period the players observe the actions chosen by all players including their own in all previous periods and only then simultaneously choose their actions for the new game Thus the important thing to remember is that players can perfectly monitor all the actions chosen in earlier periods prior to choosing an action in a subsequent period The data collected by perfectly monitoring the actions played in each period is called a history at a period To define the players strategies under a repeated game we now wish to modify Definition 28 to repeated games Definition 211 1 A period τ history of a game Hr is the list of outcomes played in all periods t 12 τ 1 2 A strategy of a player in a game repeated T times is a list of actions that the player takes in each period t t 1 2 T where each period t action is based on the period t history of the game ie maps a history Ht to an action in the set Ai Hence a strategy of a player in a repeated game is a list of actions to be played in each period τ where each period τ action of player i is based on the observed list of actions played by all players in all periods t 1 2 τ 1 summarized by the history Hτ Therefore an outcome of a repeated game would be a list of actions each player is taking in every period whereas the period τ payoff to each player is a function of the actions played by the players in period τ Consider our PeaceWar game described in Table 21 and suppose that this game is repeated T times in periods 1 2 T where T is Page 29 an integer number satisfying We denote by 0 ρ 1 the time discount parameter for each player That is the parameter ρ is the present value of one dollar to be received by the player next period Another way of interpreting ρ is to assume that our players live in a world with perfect capital markets where players can lend or borrow any amount of money at a given real interest r r 0 In this case we can assume that the economys real interest r would adjust so that r 1ρ 1 or ρ 11 r We now make the following assumption regarding the players payoffs in a repeated game Assumption 21 Let denote the action taken by player i in period t i 1 2 t 1 2 T Also let be the period t payoff to player i i 1 2 where is given in Table 21 Then the payoff to player i when the game is repeated T times is defined by If the number of players is greater than two then replace with at where We distinguish between two types of repeated games a finitely repeated game and an infinitely repeated game 231 Finitely repeated game Suppose that the PeaceWar game is repeated T times in periods 1 2 T where T is a finite integer number satisfying In Section 21 we have shown that WAR WAR is a unique NE for this oneshot game Now suppose that the game is played twice in two consecutive periods T 2 If we apply Definition 211 what strategies are available to say country 1 More precisely how many strategies are there in country 1s strategy set Claim 25 There are 32 25 available strategies to country I in this twoaction twoperiod repeated game Proof Let us first look at the second period In the second period there could be four possible histories resulting from the four possible firstperiod lists of players actions That is period 2 history satisfies Page 30 In the second period there are two possible actions country 1 can take WAR and PEACE Now in order to fully specify a strategy country 1 has to specify which action will be taken for every possible history Hence the number of secondperiod actions is 24 On top of this there are two possible actions available to country 1 in period 1 Hence the number of strategies available to country 1 in a twoaction twoperiod repeated game is 2 24 25 Similarly if the game is repeated three times T 3 the strategy set of country 1 contains strategies since in the third period there are 16 4 x 4 possible histories resulting from four possible lists of players actions in each period We now state our main proposition for finitely repeated games Proposition 24 For any finite integer T the Ttimes repeated PeaceWar game has a unique SPE where each country plays WAR in each period Thus Proposition 24 states that no matter how many times the PeaceWar game is repeated it could be one or it could be a billion times the unique SPE is WAR played by all players in every period Proof Using backward induction let us suppose that the countries have already played in T1 periods and that now they are ready to play the final Ts period game Then since period T is the last period that the game is played the Ts period game is identical to the single oneshot PeaceWar game Hence a unique NE for the Ts period game is WAR played by each country Now consider the game played in period T 1 Both players know that after this game is completed they will have one last game to play in which they both will not cooperate and play WAR Hence in T 1 each player would play the dominant strategy WAR Working backwards in each of the proceeding periods T2 T3 until period 1 we can establish that WAR will be played by every player in each period 232 Infinitely repeated game Now suppose that the game is repeated infinitely many times ie The difference between the infinitely repeated game and the small or large but finitely repeated game is that in an infinitely repeated game backward induction used in the proof of Proposition 24 cannot be used to arrive at equilibrium outcomes since there is no final period to start the backward induction process Page 31 The trigger strategy We restrict the discussion of strategy in infinitely repeated games to one type called trigger strategies In the class of trigger strategies each player cooperates in period t playing as long as all players cooperated in period τ 1 However if any player did not cooperate and played WAR in period τ 1 then player i pulls the trigger and plays the noncooperative action forever That is for every t τ τ 1 τ 2 Formally Definition 212 Player i is said to be playing a trigger strategy if for every period τ τ 1 2 That is country i cooperates by playing PEACE as long as no country including itself deviates from the cooperative outcome However in the event that a country deviates even once country i punishes the deviator by engaging in a WAR forever Equilibrium in trigger strategies We now seek to investigate under what conditions the outcome where both countries play their trigger strategies constitutes a SPE Proposition 25 If the discount factor is sufficiently large then the Outcome where the players play their trigger strategies is a SPE Formally trigger strategies constitute a SPE if ρ 12 Proof Let us look at a representative period call it period τ and suppose that country 2 has not deviated in periods 1 τ Then if country 1 deviates and plays Table 21 shows that However given that country 1 deviates country 2 would deviate in all subsequent periods and play for every since country 2 plays a trigger strategy Hence from period τ 1 and on country 1 earns a payoff of 1 each period Therefore the period τ 1 sum of discounted payoffs to country 1 for all periods Note that we used the familiar formula for calculating the present value of an infinite stream of payoffs given by Hence if country 1 deviates in period τ its sum of discounted payoffs is the sum period τs payoff from playing WAR while country 2 plays PEACE equal to plus the discounted infinite sum of payoffs when both countries play WAR sum of discounted payoffs of 1 each period Thus if country 1 deviates from PEACE in period τ then Page 32 However if country 1 does not deviate then both countries play PEACE indefinitely since country 2 plays a trigger strategy Hence both countries gain a payoff of 2 each period Thus Comparing 23 with 24 yields the conclusion that deviation is not beneficial for country 1 if ρ 12 Since no unilateral deviation is beneficial to any country at any subgame starting at an arbitrary period τ we conclude that no unilateral is beneficial to a country at any period t So far we have showed that when both countries play the trigger strategy no country has the incentive to unilaterally deviate from playing PEACE In the language of game theorists we showed that deviation from the equilibrium path is not beneficial to any country However to prove that the trigger strategies constitute a SPE we need to show that if one country deviates and plays WAR the other country would adhere to its trigger strategy and would play WAR forever In the language of game theorists to prove SPE we need to prove that no player has the incentive to deviate from the played strategy even if the game proceeds off the equilibrium path To prove that note that if country 1 deviates from PEACE in period τ then Definition 212 implies that country 1 will play WAR forever since Definition 212 states that any deviation by country 1 or country 2 would trigger country 1 to play WAR forever Hence country 2 would punish country 1 by playing WAR forever since WAR yields a payoff to country 2 of 1 each period compared with payoff of 0 if country 2 continues playing PEACE Altogether the trigger strategies defined in Definition 212 constitute a SPE for the infinitely repeated PeaceWar game Proposition 25 demonstrates the relationship between the players time discount factor given by ρ and their incentive to deviate from the cooperative action That is when players have a low discount factor say ρ is close to zero the players do not care much about future payoffs Hence cooperation cannot be a SPE since the players wish to maximize only their first period profit However when ρ is large ρ 12 in our case players do not heavily discount future payoffs so cooperation becomes more beneficial to the players since the punishment on deviation becomes significant because the discounted flow of payoffs under cooperation 2 per period is higher than the shortrun gain from Page 33 deviation a payoff of 3 for one period and 1 thereafter This discussion leads to the following corollary Corollary 21 In an infinitely repeated game cooperation is easier to sustain when players have a higher time discount factor 233 A discussion of repeated games and cooperation In this section we have shown that a oneshot game with a unique noncooperative Nash equilibrium can have a cooperative SPE when it is repeated infinitely However note that in the repeated game this SPE is not unique For example it is easy to show that the noncooperative outcome where each country plays WAR in every period constitutes a SPE also Moreover the Folk Theorem Folk because it was well known to game theorists long before it was formalized states that for a sufficiently high time discount factor a large number of outcomes in the repeated game can be supported as a SPE Thus the fact that we merely show that cooperation is a SPE is insufficient to conclude that a game of this type will always end up with cooperation All that we managed to show is that cooperation is a possible SPE in an infinitely repeated game Finally let us look at an experiment Robert Axelrod conducted in which he invited people to write computer programs that play the Prisoners Dilemma game against other computer programs a large number of times The winner was the programmer who managed to score the largest sum over all the games played against all other programs The important result of this tournament was that the program that used a strategy called TitforTat won the highest score The TitforTat strategy is different from the trigger strategy defined in Definition 212 because it contains a less severe punishment in case of deviation In the TitforTat strategy a player would play in period t what the opponent played in period t 1 Thus even if deviation occurred once the opponent resumes cooperation the players would switch to cooperation in a subsequent period Under the trigger strategy once one of the players deviates the game enters a noncooperative phase forever 24 Appendix Games with Mixed Actions The tools developed in this appendix are not implemented in this book and are brought up here only for the sake of completeness Thus this appendix is not necessary to study this book successfully and the beginning readers are urged to skip this appendix Games with mixed actions are those in which the players randomize over the actions available in their action sets Often it is hard to Page 34 motivate games with mixed actions in economics modeling This is not because we think that players do not choose actions randomly in real life On the contrary the reader can probably recall many instances in which he or she decided to randomize actions The major reason why games with mixed actions are hard to interpret is that it is not always clear why the players benefit from randomizing among their pure actions The attractive feature of games with mixed actions is that a Nash equilibrium in mixed actions always exists Recall that Proposition 22 demonstrates that a Nash equilibrium in pure actions need not always exist In what follows our analysis will concentrate on the TopBottomLeftRight given in Table 25 The reason for focusing on the game in Ms β Lleft Rright Ms α Ttop 0 0 0 1 Bbottom 1 0 1 3 Table 25 NE in mixed actions Table 25 is that we show that a Nash equilibrium in mixed actions exists despite the fact that a Nash equilibrium in pure actions does not the reader is urged to verify that indeed a Nash equilibrium in pure actions does not exist We now wish to modify a game with pure strategies to a game where the players choose probabilities of taking actions from their action sets Recall that by Definition 21 we need to specify three elements a the list of players already defined b the action set available to each player and c the payoff to each player at each possible outcome the payoff function for each player Definition 213 1 A mixed action of player α is a probability distribution over playing aα T and playing aα B Formally a mixed action of player α is a probability τ such that player α plays T with probability τ and plays B with probability 1 τ 2 A mixed action of player β is a probability λ such that player β plays L with probability λ and plays R with probability 1 λ 3 An action profile of a mixed actions game is a list τ λ ie the list of the mixed action chosen by each player Page 35 4 An outcome of a game with mixed actions is the list of the realization of the actions played by each player Definition 213 implies that the mixedaction set of each player is the interval 01 where player α picks a and player β picks a The reader has probably noticed that Definition 213 introduces a new term action profile which replaces the term outcome used in normal form games Definition 21 The reason for introducing this term is that in a game with mixed actions the players choose only probabilities for playing their strategies so the outcome itself is random In games with pure actions the term action profile and the term outcome mean the same thing since there is no uncertainty However in games with mixed actions the term action profile is used to describe the list of probability distributions over actions chosen by each player whereas the term outcome specifies the list of actions played by each player after the uncertainty is resolved Our definition of the mixed extension of the game is incomplete unless we specify the payoff to each player under all possible action profiles Definition 214 A payoff function of a player in the mixedaction game is the expected value of the payoffs of the player in the game with the pure actions Formally for any given action profile λ τ the expected payoff to player i i α β is given by According to Definition 21 our game is now well defined since we specified the action sets and the payoff functions defined over all possible action profiles of the mixed actions game Applying the NE concept defined in Definition 24 to our mixedactions game we can state the following definition Definition 215 An action profile where is said to be a Nash equilibrium in mixed actions if no player would find it beneficial to deviate from her or his mixed action given that the other player does not deviate from her or his mixed action Formally Page 36 We now turn to solving for the Nash equilibrium of the mixedactions extension game of the game described in Table 25 Substituting the payoffs associated with the pure outcomes of the game in Table 25 into the mixed payoff functions given in Definition 214 yields and Restating Definition 215 we look for a pair of probabilities that satisfy two conditions a for a given maximizes given in 26 and b for a given maximizes given in 27 It is easy to check that the players payoffs 26 and 27 yield bestresponse functions see Definition 25 given by That is when player β plays R with a high probability 1 λ 12 player αs best response is to play T with probability 1 τ 1 in order to minimize the probability of getting a payoff of 1 However when player β plays L with a high probability λ 12 player αs best response is to play B with probability 1 τ 0 in order to maximize the probability of getting a payoff of 1 Similar explanation applies to the bestresponse function of player β The bestresponse functions of each player are drawn in Figure 23 Equations 28 and Figure 23 show that when β plays λ 12 player α is indifferent to the choice among all her actions That is when λ 12 the payoff of player α is the same zero for every mixed action In particular player α is indifferent to the choice between playing a pure strategy meaning that τ 0 or τ 1 and playing any other mixed actions 0 τ 1 Similarly player β is indifferent to the choice among all her mixed actions when player α plays τ 34 Although a NE in pure actions does not exist for the game described in Table 25 the following proposition shows Proposition 26 There exists a unique NE in mixed actions for the game described in Table 25 In this equilibrium τ 34 and λ 12 Page 37 Figure 23 Bestresponse functions for the mixedaction extended game The proposition follows directly from the righthand side of Figure 23 that shows that the two best response functions given in 28 have a unique intersection Finally the bestresponse functions given in 28 have a property of being composed of horizontal or vertical line segments Since the equilibrium occurs when the two curves intersect in their middle sections we have it that under the NE mixed outcome each player is indifferent to the choice among all other probabilities that can be played assuming that the other player does not deviate from the mixed action This result makes the intuitive interpretation of a mixedaction game rather difficult because there is no particular reason why each player would stick to the mixed action played under the NE 25 Appendix Games with Imperfect Information Games with imperfect information are brought up here only for the sake of completion and the beginning readers are urged to skip this appendix Games with imperfect information describe situations where some players do not always observe the action taken by another player earlier in the game thereby making the player unsure which node has been reached For example Figure 24 describes a variant of the PilotTerrorist game given in Figure 21 In Figure 24 we suppose that the terrorist cannot monitor the direction in which the pilot is flying say because the terrorist cannot read a compass or because the pilot disables some of the navigation equipment The broken line connecting nodes IIC and IIN describes an information set for the terrorist The information set tells us that in this game the terrorist cannot distinguish whether node IIC or IIN has been reached Thus when the terrorist has her turn to make a move she has to choose an action without knowing the precise node she is on Formally Page 38 Figure 24 A game with imperfect information Information sets Definition 216 An information set for a player is a collection of nodes in which the player has to choose an action When a player reaches an information set the player knows that the particular information set has been reached but if the information set contains more than one node the player does not know which particular node in this collection has been reached We now have the tools to define a game with imperfect information Definition 217 An extensive form game is called 1 A game with imperfect information if one of the information sets contains more than one node 2 A game with perfect information if each information set contains a single node Thus all the extensive form games analyzed in Section 22 are games with perfect information since each information set coincides with a single node We now slightly extend our definition of a strategy Definition 28 to incorporate games with imperfect information Definition 218 In a game with imperfect information a strategy for a player is a list of actions that a player chooses at any information set where the player is entitled to take an action Thus Definition 218 provides a more general definition of a strategy compared with Definition 28 since a strategy is a list of actions a player chooses in each information set rather than in each node where the player is entitled to take an action Under perfect information of course Definitions 28 and 218 coincide since under perfect information each information set is a singleton Finally we need to extend our definition of subgames Definition 29 to incorporate games with imperfect information Page 39 Definition 219 A subgame is an information set that contains a single node and all the subsequent decision and terminal nodes provided that all subsequent nodes are not contained in information sets containing nodes that cannot be reached from the subgame Figure 25 illustrates a game with imperfect information In Figure 25 the nodes labeled A D and G are starting nodes for a subgame However the nodes labeled B C E and F are not starting nodes for a subgame since some subsequent nodes are contained in information sets containing nodes that cannot be reached from these nodes Figure 25 Game with imperfect information Subgames For example the modified PilotTerrorist game described in Figure 24 has only one subgame which is the original game itself because all subsequent nodes are contained in information sets containing more than one node We conclude our discussion of games with imperfect information with solving for NE and SPE for the modified PilotTerrorist game described in Figure 24 First all the possible outcomes for this game are given by NY B NY NB Cuba B and Cuba NB Thus in the PilotTerrorist game under imperfect information the number of outcomes has been reduced from eight to four since the terrorist now takes a decision at one information set compared with two nodes under perfect information Second since this game does not have any proper subgames any NE is also a SPE Hence in this case the set of NE outcomes coincides with the SPE outcomes Thus we can easily conclude that NY NB constitutes both NE and SPE outcomes Page 40 26 Exercises 1 Using Definition 25 a Write down the bestresponse functions for country 1 and country 2 for the PeaceWar game described in Table 21 and decide which outcomes constitute NE b Write down the bestresponse functions for Jacob and Rachel for the game described in Table 23 and decide which outcomes constitute a NE if there are any c Write down the bestresponse functions for player 1 and player 2 for the game described in Table 25 and decide which outcomes constitute a NE if there are any 2 Consider the normal form game described in Table 26 Find the conditions on the parameters a b c d e f g and h that will ensure that Ms β Lleft Rright Ms α Ttop a b c d Bbottom e f g h Table 26 Normal form game Fill in the conditions on payoffs a the outcome T L is a NE b the outcome T L is an equilibrium in dominant actions c the outcome T L Pareto dominates all other outcomes d the outcome T L is Pareto noncomparable to the outcome B R 3 Consider the Travelers Dilemma where two travelers returning home from a remote island where they bought identical rare antiques find out that the airline has managed to smash these rare antiques The airline manager assures the passengers of adequate compensation Since the airline manager does not know the actual cost of the antiques he offers the two travelers the opportunity to write down separately on a piece of paper the true cost of the antiques which is restricted to be any number between 2 and 100 Let ni denote that value stated by traveler i i 1 2 and assume that the travelers cannot communicate with each other during this game The airline manager states the following compensation rules a If traveler i writes down a larger number than traveler j ie ni nj then he assumes that j is honest and i is lying Hence in this case the airline manager will pay ni 2 to traveler i and nj 2 to traveler j Thus the manager penalizes the traveler assumed to be lying and rewards the Page 41 one assumed to be honest b If ni nj then the manager assumes that both travelers are honest and pays them the declared value of the antiques Letting n1 and n2 be the actions of the players answer the following questions a Under Definition 26 which outcomes are Pareto Optimal b Under Definition 24 which outcomes constitute a Nash equilibrium for this game 4 Consider a normal form game between three major car producers C F and G Each producer can produce either large cars or small cars but not both That is the action set of each producer i i C F G is We denote by ai the action chosen by player i and by πiaC aF aG the profit to firm i Assume that the profit function of each player i is defined by Answer the following questions a Does there exist a Nash equilibrium when Prove your answer b Does there exist a Nash equilibrium when Prove your answer 5 Figure 26 describes an extensive form version of the Battle of the Sexes game given initially in Table 22 Work through the following problems a How many subgames are there in this game Describe and plot all the subgames b Find all the Nash equilibria in each subgame Prove your answer c Find all the subgame perfect equilibria for this game d Before Rachel makes her move she hears Jacob shouting that he intends to go to the opera ie play ω Would such a statement change the subgame perfect equilibrium outcomes Prove and explain 6 This problem refers to mixed actions games studied in the appendix section 24 Consider the Battle of the Sexes game described in Table 22 a Denote by θ the probability that Jacob goes to the OPERA and by ρ the probability that Rachel goes to the OPERA Formulate the expected payoff of each player Page 42 b Draw the bestresponse function for each player RJρ and RRθ c What is the NE in mixed actions for this game d Calculate the expected payoff to each player in this NE e How many times do the two bestresponse functions intersect Explain the difference in the number of intersections between this game and the bestresponse functions illustrated in Figure 23 Figure 26 Battle of the Sexes in extensive form 27 References Aumann R 1987 Game Theory In The New Palgrave Dictionary of Economics edited by J Eatwell M Milgate and P Newman New York The Stockton Press Axelrod R 1984 The Evolution of Cooperation New York Basic Books Binmore K 1992 Fun and Games Lexington Mass DC Heath Friedman J 1986 Game Theory with Applications to Economics New York Oxford University Press Fudenberg D and J Tirole 1991 Game Theory Cambridge Mass MIT Press Gibbons R 1992 Game Theory for Applied Economists Princeton NJ Princeton University Press McMillan J 1992 Games Strategies and Managers New York Oxford University Press Moulin H 1982 Game Theory for the Social Sciences New York New York University Press Osborne M and A Rubinstein 1994 A Course in Game Theory Cambridge Mass MIT Press Rasrnusen E 1989 Games and Information An Introduction to Game Theory Oxford Blackwell Page 43 Chapter 3 Technology Production Cost and Demand Large increases in cost with questionable increase in performance can be tolerated only for race horses and fancy spouses Lord Kelvin 18241907 President of the Royal Society This chapter reviews basic concepts of microeconomic theory Section 31 Technology and Cost introduces the singleproduct production function and the cost function Section 32 analyzes the basic properties of demand functions The reader who is familiar with these concepts and properties can skip this chapter and proceed with the study of industrial organization The student reader should note that this chapter reflects the maximum degree of technicality needed to grasp the material in this book Thus if the reader finds the material in this chapter to be comprehensible then the student should feel technically well prepared for this course 31 Technology and Cost The production function reflects the knowhow of a certain entity that we refer to as the firm This knowhow enables the firm to transform factors of production into what we call final goods In general we refrain from addressing the philosophical question of where technological knowhow comes from However in chapter 9 Research and Development we do analyze some factors that affect the advance of technological knowhow Page 44 311 The production function We assume that two inputs are needed to produce the single final good We call these inputs labor and capital Note that we restrict our discussion to production technologies for producing one and only one type of output In reality many production processes yield more than one type of output For example an oil refinery yields a variety of oil and plastic products from the same input of crude oil We postpone the discussion of multiproduct production activities to our analysis of the airline industry given in section 172 The production function represents a mapping from the amount of labor denoted by l and the amount of capital denoted by k employed in the production process to the number of units of output produced We represent this relationship by a function f where the number of units of output is given by Q flk Assuming that the function f is twice continuously differentiable with respect to both arguments we define the marginal product of labor function MPLlk as the amount of output increase associated with a small increase in the amount of labor Formally we define the marginal product of labor and capital functions by For example the marginalproduct functions associated with the class of production functions where a α β 0 are given by and It is important to note that the marginal product of a factor is a function not necessarily a constant of the amount of labor and capital used in the production process In our example meaning that in this production process the marginal product of labor gets larger and larger as the amount of labor becomes scarce So far we have not discussed the relationship between the two factors We therefore make the following definition Definition 31 1 Labor and capital are called supporting factors in a particular production process if the increase in the employment of one factor raises the marginal product of the other factor Formally if Page 45 2 Labor and capital are called substitute factors in a particular production process if the increase in the employment of one factor decreases the marginal product of the other factor Formally if In our example the reader can verify that labor and capital are supporting factors if β 1 and substitute factors if β 1 We conclude the discussion of the production function by looking at the effect of input expansion on the amount of production Formally Definition 32 Let λ be any number greater than 1 Then a production technology Q flk is said to exhibit 1 Increasing returns to scale IRS if That is if expanding the employment of labor and capital by the factor of λ will increase the output by more than a factor of λ 2 Decreasing returns to scale DRS if That is if expanding the employment of labor and capital by the factor of λ will increase the output by less than a factor of λ 3 Constant returns to scale CRS if That is if expanding the employment of labor and capital by the factor of λ will increase the output by exactly a factor of λ In our example the production technology exhibits IRS if if and only if αβ 1 312 The cost function The cost function is a mapping from the rental prices of the factors of production and the production level to the total production cost The cost function is a technological relationship that can be derived from the production function Let W denote wage rate and R the rental price for one unit of capital The cost function is denoted by the function TCW R Q measures the total production cost of producing Q units of output when factor prices are W for labor and R for capital We define the average cost function by the ratio of the total production cost to output level Formally the average cost function the Page 46 cost per unit of output at an output level Q is defined by We define the marginal cost function as the change in total cost resulting from a small increase in output level Formally the marginal cost function at an output level Q is defined by As an example consider the total cost function given by TCQ F cQ2 This cost function is illustrated on the left part of Figure 31 We refer to F as the fixed cost parameter since the fixed Figure 31 Total average and marginal cost functions cost is independent of the output level It is straightforward to calculate that ACQ FQ cQ and that MCQ 2cQ The average and marginal cost functions are drawn on the right part of Figure 31 The MCQ curve is linear and rising with Q and has a slope of 2c The ACQ curve is falling with Q as long as the output level is sufficiently small and is rising with Q for higher output levels Thus in this example the cost per unit of output reaches a minimum at an output level We now demonstrate an easy method for finding the output level that minimizes the average cost Proposition 31 If the average cost function reaches a minimum at a strictly positive output level then at that particular output level the average cost equals the marginal cost Formally if Qmin 0 minimizes ACQ then ACQmin MCQmin Proof At the output level Qmin the slope of the ACQ function must be zero Hence Page 47 Hence To demonstrate how useful Proposition 31 could be we now return to our example illustrated in Figure 31 where TCQ F cQ2 Proposition 31 states that in order to find the output level that minimizes the cost per unit all that we need to do is extract Qmin from the equation ACQmin MC Qmin In our example Hence and 313 Duality between production and cost functions We now provide a simple illustration of the relationship between production and cost functions for the case of a singleinput production function Suppose that only labor is required for producing the final good and let the production technology be given by Q fl lγ γ 0 This production function is illustrated in the upper part of Figure 32 for three parameter cases where 0 λ 1 λ 1 and λ 1 In what follows we show how the cost function can be derived from the production function Let ω denote the wage rate Now by inverting the production function we obtain l Q1λ The total cost is the wage rate multiplied by the amount of labor employed in the production process Hence TC ωl ωQ1λ which is illustrated in the middle part of Figure 32 again for the three parameter cases where 0 λ 1 λ 1 and λ 1 We conclude this discussion by looking at the relationship between the production and cost function regarding the expansion of the production activity More precisely applying Definition 32 to the production function Q lλ we have it that λlγ λlγ if and only if γ 1 Hence this production exhibits IRS when λ 1 CRS when λ 1 and DRS when λ 1 It is important to realize that since the total cost function is derived from the production function we should be able to infer from the shape of the average cost function whether the production process exhibits IRS CRS or DRS When λ 1 there are IRS The case of IRS is Page 48 Figure 32 Duality between the production and cost functions illustrated on the right side of Figure 32 Under IRS the average cost declines with the output level reflecting the fact that under IRS the cost per unit declines with a larger scale of production say because of the adoption of assembly line technology Under CRS the cost per unit is constant reflecting a technology where an increase in the output level does not alter the per unit production cost The left side of Figure 32 reflects a DRS technology where an increase in the output level raises the per unit production cost Finally recall our twoinput example where We showed that this production technology exhibits IRS if αβ 1 and DRS if αβ 1 Deriving the cost function of this production technology would take us beyond the level of this book However for the sake of illustration we state that the cost function associated with this technology is given by where φ is a nonnegative function of W and R Now in this case Then ACQ is declining with Q if 1αβ 1 0 or αβ 1 which is the condition under which the technology exhibits IBS In contrast ACQ is rising with Q Page 49 if 1αβ 1 0 or αβ 1 which is the condition under which the technology exhibits DRS 32 The Demand Function We denote by Qp the aggregate demand function for a single product where Q denotes the quantity demanded and p denotes the unit price Formally a demand function shows the maximum amount consumers are willing and able to purchase at a given market price For example we take the linear demand function given by where a and b are strictly positive constants to be estimated by the econometrician Alternatively we often use the inverse demand function pQ which expresses the maximum price consumers are willing and able to pay for a given quantity purchased Inverting the linear demand function yields pQ a bQ which is drawn in Figure 33 Note that part of the Figure 33 Inverse linear demand demand is not drawn in the figure That is for p a the inverse demand becomes vertical at Q 0 so the demand coincides with the vertical axis and for Q ab it coincides with the horizontal axis An example of nonlinear demand function is the constant elasticity demand function given by or which is drawn in Figure 34 This class of functions has some nice features which we discuss below 321 The elasticity function The elasticity function is derived from the demand function and maps the quantity purchased to a certain very useful number which we call Page 50 Figure 34 Inverse constantelasticity demand the elasticity at a point on the demand The elasticity measures how fast quantity demanded adjusts to a small change in price Formally we define the demand price elasticity by Definition 33 At a given quantity level Q the demand is called 1 elastic if 2 inelastic if 3 and has a unit elasticity if For example in the linear case Hence the demand has a unit elasticity when Q a2b Therefore the demand is elastic when Q a2b and is inelastic when Q a2b Figure 33 illustrates the elasticity regions for the linear demand case For the constantelasticity demand function we have it that Hence the elasticity is constant given by the power of the price variable in demand function If this demand function has a unit elasticity at all output levels 322 The marginal revenue function The inverse demand function shows the maximum amount a consumer is willing to pay per unit of consumption at a given quantity of purchase The totalrevenue function shows the amount of revenue collected by sellers associated with each pricequantity combination Formally we Page 51 define the totalrevenue function as the product of the price and quantity For the linear case TRQ aQ bQ2 and for the constant elasticity demand Note that a more suitable name for the revenue function would be to call it the total expenditure function since we actually refer to consumer expenditure rather than producers revenue That is consumers expenditure need not equal producers revenue for example when taxes are levied on consumption Thus the total revenue function measures how much consumers spend at every given market price and not necessarily the revenue collected by producers The marginalrevenue function again more appropriately termed the marginal expenditure shows the amount by which total revenue increases when the consumers slightly increase the amount they buy Formally we define the marginalrevenue function by For the linear demand case we can state the following Proposition 32 If the demand function is linear then the marginalrevenue function is also linear has the same intercept as the demand but has twice the negative slope Formally MRQ a 2bQ Proof The marginalrevenue function for the linear case is drawn in Figure 33 The marginalrevenue curve hits zero at an output level of Q a2b Note that a monopoly studied in chapter 5 will never produce an output level larger than Q a2b where the marginal revenue is negative since in this case revenue could be raised with a decrease in output sold to consumers For the constantelasticity demand we do not draw the corresponding marginalrevenue function However we consider one special case where In this case p aQ1 and TRQ a which is a constant Hence MRQ 0 You have probably already noticed that the demand elasticity and the marginalrevenue functions are related That is Figure 33 shows that MRQ 0 when ηpQ 1 and MRQ 0 when ηpQ 1 The complete relationship is given in the following proposition Proposition 33 Page 52 Proof 323 Consumer surplus We conclude our discussion of the demand structure by a gross approximation of consumers welfare associated with trade We define a measure that approximates the utility gained by consumers when they are allowed to buy a product at the ongoing market price That is suppose that initially consumers are prohibited from buying a certain product Suppose next that the consumers are allowed to buy the product at the ongoing market price The welfare measure that approximates the welfare gain associated with the opening of this market is what we call consumer surplus and we denote it by CS In what follows we discuss a common procedure used to approximate consumers gain from buying by focusing the analysis on linear demand functions Additional motivation for the concept developed in this section is given in the appendix section 33 Figure 35 illustrates how to calculate the consumer surplus assuming that the market price is p Figure 35 Consumers surplus For a given market price p the consumer surplus is defined by the area beneath the demand curve above the market price Formally denoting by CSp the consumers surplus when the market price is p we define Page 53 Note that CSp must always increase when the market price is reduced reflecting the fact that consumers welfare increases when the market price falls In industrial organization theory and in most partial equilibrium analyses in economics it is common to use the consumers surplus as a measure for the consumers gain from trade that is to measure the gains from buying the quantity demanded at a given market price compared with not buying at all However the reader should bear in mind that this measure is only an approximation and holds true only if consumers have the socalled quasilinear utility function analyzed in the appendix section 33 33 Appendix Consumer Surplus The QuasiLinear Utility Case The analysis performed in this appendix is brought up here only for the sake of completeness quasi linear utility is used only once in this book in section 131 where we analyze twopart tariffs We therefore advise the beginning student to skip this appendix In this appendix we demonstrate that when consumer preferences are characterized by a class of utility functions called quasilinear utility function the measure of consumer surplus defined in subsection 323 equals exactly the total utility consumers gain from buying in the market Consider a consumer who has preferences for two items money m and the consumption level Q of a certain product which he can buy at a price of p per unit Specifically let the consumers utility function be given by Now suppose that the consumer is endowed with a fixed income of I to be spent on the product or to be kept by the consumer Then if the consumer buys Q units of this product he spends pQ on the product and retains an amount of money equals to m I pQ Substituting into 34 our consumer wishes to choose a productconsumption level Q to maximize The firstorder condition is given by and the second order by which constitutes a sufficient condition for a maximum The firstorder condition for a quasilinear utility maximization yields the inverse demand function derived from this utility function which is Page 54 given by Thus the demand derived from a quasilinear utility function is a constant elasticity demand function illustrated earlier in Figure 34 and is also drawn in Figure 36 Figure 36 Inverse demand generated from a quasilinear utility function The shaded area in Figure 36 corresponds to what we call consumer surplus in subsection 323 The purpose of this appendix is to demonstrate the following proposition Proposition 34 If a demand function is generated from a quasilinear utility function then the area marked by CS p in Figure 36 measures exactly the utility the consumer gains from consuming Q0 units of the product at a market price p0 Proof The area Csp in Figure 36 is calculated by 34 Exercises 1 Consider the CobbDouglas production function given by Q lαkβ where α β 0 Page 55 a For which values of the parameters α and β does this production technology exhibit IRS CRS and DRS b Using Definition 31 infer whether labor and capital are supporting or substitute factors of production 2 Consider the production function given by Q lα kα where α 0 a For which values of a does this production technology exhibit IRS CRS and DRS b Using Definition 31 infer whether labor and capital are supporting or substitute factors of production 3 Does the production function given by exhibit IRS CRS or DRS Prove your answer 4 Consider the cost function where Fc 0 a Calculate and plot the TCQ ACQ and MCQ b At what output level is the average cost minimized c Infer whether this technology exhibits IRS CRS or DRS Explain 5 Consider the demand function Q 99 p a At what output level does the elasticity equal 2 b At what output level does the elasticity equal 1 c Calculate and draw the marginalrevenue function associated with this demand d At what output level does the marginal revenue equal zero e Calculate the consumers surplus when p 33 and p 66 6 Consider the constantelasticity demand function where A a Solve for the inverse demand function pQ b Using 32 calculate the demand price elasticity c For what values of is the demand elastic For what values is the demand inelastic d Using Proposition 33 show that the ratio of the marginalrevenue function to the inverse demand function pQMRQ is independent of the output level Q Page 57 PART II MARKET STRUCTURES AND ORGANIZATION Page 59 We define market structure as a description of the firms behavior in a given industry or market The list of items defining firms behavior include precise specifications of 1 The actions available to each firm eg choosing a price setting quantity produced setting production capacity or location etc 2 The number of firms in the industry and whether this number is fixed or whether free entry of new firms is allowed 3 Firms expectation about the actions available to competing firms and how the competing firms will respond to each firms action 4 Firms expectation about the number of firms and potential entry Thus specifying a market Structure is similar to specifying the rules of the game or rules for interaction among existing or potentially entering new firms In many cases specifying a market structure is similar to defining a game according to Definition 21 on page 13 Figure II1 on page 61 lists most of the market structures used in this book The top of the tree in Figure II1 shows that market structures are classified into two categories competitive and imperfectly competitive The competitive market structure studied in chapter 4 and which you have probably studied in your intermediate microeconomic class assumes that each firms action set is its production quantity while each firm takes the market price as given where the market price is determined by the intersection of the market demand curve and the industrys aggregate supply curve Competitive market structures can be solved for by assuming either a fixed number of firms sometimes referred to as shortrun equilibrium or free entry sometimes referred to as a longrun equilibrium Among the imperfectly competitive market structures the reader is probably most familiar with the monopoly market structure which is studied in chapter 5 Under this market structure there is only one seller who can choose any priceoutput combination on the consumers aggregate demand curve Given the onetoone relationship between price and quantity implied by the market demand curve the monopoly is restricted to choosing a price or a quantity produced but not both Monopoly market structures can be classified as static where the monopoly sells its product only once or dynamic where the monopoly sells durable or nondurable goods over more than one period Monopoly market structures are then classified into discriminating and nondiscriminating monopolies A discriminating monopoly can earn a higher profit than a nondiscriminating one by selling the product to different consumers at different prices The duopoly two sellers and the oligopoly more than two sellers market structures are classified as cooperative and noncooperative Cooperative behavior is defined by firms colluding by agreeing to produce in total the monopolys profitmaximizing output level or to charge the Page 60 monopolys price A noncooperative behavior can be modeled either using oneshot games where all firms choose their strategic variables quantity produced or price once and at the same time or dynamically where the firms move in sequence Whether firms move simultaneously or whether they move in sequence firms choose either prices Bertrand or quantity produced Cournot Finally one market structure that economists tend to focus on assume that firms are engaged in a repeated interaction of a simultaneousmove oligopoly game That is in each period each firm chooses its action from the same action set after observing what actions have been chosen in earlier periods The upward arrow in Figure II1 hints that a fascinating possible outcome of an infinitely repeated oligopoly game is where firms choose to play their collusive cooperative actions output level or price Page 61 Figure II1 Commonly assumed and used market structures Note Ddiscriminating NDnondiscriminating Page 63 Chapter 4 Perfect Competition In perfect markets whether monopolistic or competitive price is hardly a matter of judgment and where there is no judgment there is no policy Edward S Mason Price and Production Policies of LargeScale Enterprise This chapter describes perfectly competitive markets We first need to define what do we mean by the term competitive market or equivalently a perfectly competitive market We define a competitive market or perfect competition as a market where agents buyers and sellers behave competitively But what do we mean by competitive behavior In economics the following definition is commonly used for competitive behavior Definition 41 A buyer or a seller agent in what follows is said to be competitive or alternatively to behave competitively if the agent assumes or believes that the market price is given and that the agents actions do not influence the market price Thus the assumption of competitive behavior relates only to what agents believe about the consequences of their actions That is competitive behavior implies that agents think that their actions say quantityproduced will not have any effect on the market price It is important to note that the assumption of competitive behavior is independent of how many firms or consumers there are in the market it relates only to beliefs More precisely assuming competitive behavior does not imply that the number of sellers is large In fact one of the exercises accompanying this discussion asks you to define and solve for the competitive equilibrium price when there is only one seller in the mar Page 64 ket Thus as long as the agents behave competitively the competitive equilibrium price can be solved for any number of buyers and sellers The common mixup between the assumption of competitive behavior and the assumption that the number of sellers must be large stems from two reasons First the assumption of pricetaking behavior seems more reasonable when the number of firms is large and each firm sells a small amount relative to the aggregate industry sales Second the equilibrium price solutions for some imperfectly competitive market structures converge on get closer to the competitive price when the number of firms increases Therefore when there is a large number of sellers equilibrium price under various market structures gets closer to the price solved by competitive behavior nevertheless the definition of competitive behavior is completely independent of the number of firms Suppose that our consumers demand a homogeneous product Denoting the price of the product by p and the aggregate quantity demanded by Q we assume that consumers aggregate inverse demand function is linear and is given by 41 NonIncreasing Returns to Scale Suppose that there are two firms named firm 1 and firm 2 producing this homogeneous product We denote by qi the quantity produced by firm i and by TCiqi the total cost function of firm i i 1 2 To be more specific let us assume that the firms have constant returns to scale technologies summarized by linear cost functions given by The linear cost functions have the property that the marginal cost the increment in cost due to a small increase in the production level equals the average cost cost per unit of production Formally c1 and c2 are called constant unit costs of production if ci satisfies In general constant unit costs are associated with constantreturnstoscale CRS production functions since CRS production functions represent technologies where doubling the inputs would double the output constant unit costs mean that doubling the output will exactly double the total cost of production Observe that in equation 42 we assumed with no loss of generality that firm 2 has a higher unit production cost than firm 1 or an equal Page 65 one Figure 41 illustrates the demand and the unit costs in the priceoutput space Figure 41 Competitive equilibrium under constant returns to scale Now that the economy is well defined we define a competitive equilibrium as a vector of quantities produced and a price such that 1 each firm chooses its profitmaximizing output at the given equilibrium price and 2 at the equilibrium price aggregate quantity demanded equals aggregate quantity supplied Formally Definition 42 The triplet is called a competitive equilibrium if 1 given pe solves 2 Now that we have defined competitive equilibrium we seek to solve for this equilibrium for the industry described in 41 and 42 The first step would be to calculate the supply functions of the two firms which are found from the profitmaximization procedure defined in part 1 of Definition 42 Lemma 41 The supply functions are given by Page 66 Proof Since each firm i treats p as a constant the firms profit margin defined by p ci is constant Hence p ci treated by the firm as the constant perunit profit loss if negative Therefore if p ci0 the firm would produce and if p ci 0 the firm would produce qi 0 whereas if p ci 0 the firm makes a zero profit at every level of production implying that the output level is indeterminate We search for the equilibrium price that would satisfy Definition 42 However observing 44 can tell us which prices cannot constitute an equilibrium More specifically any price above the unit cost of firm 1 p c1 cannot be an equilibrium price since 44 tells us that if p c1 however the demand function 41 tells us that the quantity demanded at any price is always finite Hence for p c1 the quantity supplied exceeds the quantity demanded thereby violating part 2 of Definition 42 Thus if a competitive equilibrium exists it must be that However if the supply functions 44 imply that q1 q2 0 and since the quantity demanded is greater than zero it exceeds the quantity supplied thereby violating part 2 of Definition 42 Hence Proposition 41 If the unique competitive equilibrium price is pe c1 and 1 if c2 c1 firm 2 is not producing and 2 if c2 c1 then and q1 That is the aggregate industry output level is determined but the division of the industry output between the firms is indeterminate Finally let us make three remarks a Observe that if a c1 meaning that the demand is low then neither firm would produce b This model can be easily extended to any number of firms Clearly in equilibrium only the firms with the lowest unit cost would produce c Definition 42 allows us to impose the competitive market structure even if there is only one firm For example if there is only one firm with a unit cost then pe c and constitute a unique competitive equilibrium 42 Increasing Returns to Scale The analysis in subsection 41 is valid only if firms technologies exhibit decreasing or constant returns to scale technologies Suppose now that firms have increasingreturnstoscale IRS technologies To simplify we assume that there is only one firm whose total cost of production is composed of a fixed cost independent of the production level and a Page 67 constant marginal cost Formally the total cost of producing q units of output is given by Figure 42 illustrates the marginal and averagetotalcost functions associated with this technology showing that the average cost decreases and approaches the constant marginal cost as the output level increases since the average fixed cost approaches zero Figure 42 Decreasing average cost technology Our main result is given in the following proposition Proposition 42 Let a c If firms technologies exhibit increasing returns to scale decreasing average cost a competitive equilibrium does not exist Proof By a way of contradiction suppose that a competitive equilibrium exists Then from Figure 42 the equilibrium price has to satisfy one of the following or pe c a Suppose that Then for every q 0 That is the equilibrium price is below that average cost for all strictly positive output levels Hence the firm would produce qe 0 But qe 0 cannot be an equilibrium since at this price range the quantity demanded is strictly positive and excess demand violates part 2 of Definition 42 b Now suppose that Then for q exceeding a certain level That is the equilibrium price is above the average cost for sufficiently large output levels Moreover the perunit profit measured by increases with q implying that the competitive firm produces But cannot be an equilibrium the quantity demanded is always finite and excess supply violates part 2 of Definition 42 Page 68 43 MarginalCost Pricing and Social Welfare In this section we demonstrate a very important feature of the competitiveequilibrium outcome More precisely in this section we demonstrate that the perfectly competitive market structure yields a market outcome that maximizes social welfare to be defined below We first wish to define a social welfare function for our economy In subsection 323 on page 52 we defined the concept of consumer surplus denoted by CSp and showed that this measure approximates consumers utility level at a given market price In order to fully capture the economys welfare we also need to take into consideration the fact that firms are owned by our consumers and therefore we defined social welfare by the sum of consumer surplus and firms profits Formally Definition 43 Let the market price be given by p and suppose that there are firms in the industry We define social welfare by In what follows we show that the perfectly competitive market structure yields a market price that maximizes social welfare as defined in Definition 43 Indeed we are going to prove something more general than that We will show that when the market price equals the marginal cost of producing firms then the quantity produced and consumed maximizes social welfare Now given that competitive equilibrium results in marginalcost pricing it is clear the competitive outcome maximizes welfare Figure 43 illustrates the welfare level for every given market price Figure 43 illustrates three important areas under the inverse demand curve if we assume that the market price is p0 0 The consumer surplus defined in subsection 323 is given by CSp0 α The industry profit is the distance between price and unit cost multiplied by the quantity sold and is therefore given by Σπp0 β By definition the total welfare is given by W α β Figure 43 shows that the area marked by γ is not part of measuring welfare Indeed the area measured by γ is considered to be the deadweight loss associated with higherthanmarginalcost pricing The intuition behind the definition of the q loss area is that since the demand function slopes downward a higherthanmarginalcost price would reduce the quantity demanded The consumer surplus loss associated with a lower consumption level cannot be fully captured by a higher profit level if any associated with a higher price Page 69 Figure 43 Marginalcost pricing and social welfare CSp α Σπip β Wp α β Figure 43 shows that when the market price is reduced from p0 to p c the deadweightloss area merges into the consumer surplus In addition the reduction in industry profit is offset by the increase in consumer surplus Altogether social welfare increases with a price reduction as long as price exceeds marginal costs Finally notice that we do not discuss cases where market prices are below unit costs p c since when the price is reduced below marginal cost the increase in firms loss exceeds the increase in consumer surplus 44 Exercises The market demand curve for a certain product is given by Qp 120 p where p is the market price and Q denotes the quantity purchased by the consumers Suppose that the product is produced with a single factor of production called labor denoted by L Assume that each firm i can hire any amount of labor at a fixed given wage rate denoted by ω 0 The production function of each firm i is given by where Li is the amount of labor employed by firm i 1 Suppose that there is only one firm producing this product call it firm 1 Solve the firms profit maximization problem and prove that the firms supply curve is given by 2 Suppose now that ω 1 Using Definition 42 solve for the competitive equilibrium price and quantity for this singlefirm industry 3 Calculate the profit of this firm in a competitive equilibrium Page 70 4 Now suppose that there are two firms whose output levels are denoted by q1 and q2 Solve for the competitive equilibrium price and quantities produced by each firm 5 Compare the market price and aggregate production when the competitive equilibrium is solved for a single firm and when it is solved for a twofirm industry 6 Draw the supply curve of each firm and then plot the aggregate industrysupply curve Label production on the horizontal axis and price on the vertical axis Then draw the industrys demand curve and graphically solve for the competitiveequilibrium price 45 References Mason E 1939 Price and Production Policies of LargeScale Enterprise American Economic Review 29 pt 2 6174 Page 71 Chapter 5 The Monopoly Every person who shall monopolize or attempt to monopolize or combine and conspire with any other person or persons to monopolize any part of the trade or commerce shall be deemed guilty of a felony Sherman Antitrust Act of 1890 In this chapter we develop a theory of a single seller facing competitive pricetaking consumers in one or several markets over one or several periods It is important to fully understand the extreme monopoly case since when few firms compete the firms can always exercise some monopoly power In addition for the sake of simplicity several arguments in this book are demonstrated only for the monopoly case rather than for some other forms of market structures A single seller is facing a downward sloping demand curve Thus since consumers are always on their demand curve the monopoly can determine either the price for the product or the quantity supplied That is a decision about price implies a decision about quantity produced and vice versa since quantity and price are related via the demand curve For this reason the monopoly needs to devote resources to the careful study of the demand curve facing its product that is the monopoly has to familiarize itself with all the demand properties discussed in section 32 After estimating the demand curve the monopoly has to study the market demand to determine its profitmaximizing output Section 51 presents the familiar monopoly profitmaximization problem for a single market Section 52 Monopoly and Welfare reviews the standard welfare argument demonstrating the welfare loss associated with a lowerthanoptimal production level Section 53 Discriminating Monopoly departs from the singlemarket assumption and analyzes a profitmaximizing monopoly that can charge different prices in different Page 72 markets Section 54 The Cartel and the Multiplant Monopoly analyzes two forms of collusive contractual arrangements among all the firms producing in the industry that together behave as a monopoly profit maximizing entity Section 55 Durable Good Monopolies analyzes the monopolys behavior over a period of time where the monopoly sells a good that provides services for more than one period The appendix section 56 discusses the legal antitrust approach to the monopoly and to price discrimination 51 The Monopolys ProfitMaximization Problem The technology of the firm is summarized by its cost function which relates the quantity produced to the cost of producing this quantity Let TCQ denote the total cost function of the monopoly Denoting by πQ the monopolys profit level when producing Q units of output the monopoly chooses Qm to A necessary but not sufficient condition for Qm 0 to be the monopolys profitmaximizing output is Notice that 51 is only a necessary condition meaning that if the profit maximizing output is strictly positive then it has to satisfy 51 However especially if the monopoly has to pay high fixed costs it is possible that the monopolys profitmaximizing output level is Qm 0 Altogether 51 implies that if a profitmaximizing monopoly produces a strictly positive output level Qm then the profit output level must satisfy the condition MRQm MCQm Thus the easiest method for finding the monopolys profitmaximizing output level is first to solve for Qm from 51 and then to substitute it into the total profit function to check whether πQm is greater than or equal to zero If it is not then the monopoly sets Qm 0 and if profit is nonnegative then the output level solved from 51 is the profitmaximizing output level After finding the monopolys profitmaximizing output the price charged by the monopoly can be found by substituting Qm into the demand function Figure 51 illustrates the monopoly solution for the case where TCQ F cQ2 and a linear demand function given by pQ a bQ Figure 51 left shows the case where the demand is high enough or the fixed cost is low enough so that the monopoly Page 73 Figure 51 The monopolys profit maximizing output would produce Qm 0 and hence would charge a price of pm Figure 51 right illustrates a case where the demand is so low that the monopolys price cannot cover the average cost Hence Qm 0 To solve it explicitly note that by Proposition 32 we have it that MRQ a 2bQ Hence if Qm 0 then by 51 Qm solves a 2bQm 2cQm implying that Consequently Altogether the monopolys profitmaximizing output is given by 52 Monopoly and Social Welfare The US legal system discourages monopolies see the appendix subsection 561 In what follows we provide two arguments for why monopolies are discouraged 521 The conventional argument against a monopoly Figure 52 illustrates the conventional argument against monopolies The monopoly equilibrium pm Qm is illustrated in the left side of Page 74 Figure 52 Monopoly and social welfare Figure 52 where the area CS measures the consumers surplus see subsection 323 We define the total welfare W as the sum of industry profit and consumers surplus Formally which is measured by the entire shaded area of the left side of Figure 52 The tight side of Figure 52 illustrates a welfareimproving case involving marginalcost pricing associated with perfectly competitive markets see section 43 on page 68 Comparing the monopoly outcome with the marginalcost pricing outcome reveals that whereas the industrys profit is lower under marginalcost pricing possibly zero the CS is clearly much larger under marginal cost pricing That is the gain to total welfare when the market outcome changes from monopoly to perfect competition is precisely the deadweightloss area marked by DL associated with the monopoly market structure 522 The social cost of a monopoly Posner 1975 argued that the cost to the society associated with the existence of a monopoly is much higher than the deadweightloss area marked by DL in Figure 52 That is following Tullock 1967 he argued that the pursuit of monopoly rents is itself a competitive activity and one that consumes resources This activity was given the term rent seeking by Krueger 1974 More precisely Tullock and Posner argued that the social cost of having a monopoly should also include the costs of deterring competition that are analyzed in section 83 and in Section 84 The point is that firms wishing to obtain a monopoly status or wishing to maintaining a monopoly position must allocate resources for that goal These resources may or may not be counted as a waste to the Page 75 economy Resources allocated to establishing or maintaining monopoly power that should not be considered as reducing welfare include 1 RD leading to a patent monopoly right for seventeen years see section 94 since the RD improves technologies and results in new products 2 Bribes to politicians or civil servants for the purpose of getting exclusive business rights since this constitutes only a transfer of wealth Now resources allocated to the establishment of monopoly power that may count as social waste include 1 Persuasive advertising see section 111 needed to convince consumers that alternative brands are inferior 2 Resources needed to preempt potential entrants from entering the industry Also excessive production or investment in capital for the purpose of making entry unprofitable for potential competitors see section 83 3 Lobbying costs needed to convince the legislators that a particular monopoly is not harmful provided that these costs divert resources from productive activities 4 Excessive RD resulting from a patent race 53 Discriminating Monopoly Our analysis so far has focused on monopolies charging a single uniform price to all customers A firm can however increase its profit by charging different prices to consumers with different characteristics That is a firm may be able to differentiate among consumers according to tastes income age and location in order to charge consumers with different characteristics different prices Note however that in order to be able to charge consumers different prices a firm must possess the means for making arbitrage buying low for the purpose of reselling at a high price impossible In other words price discrimination is impossible when those consumers who are able to purchase at a low price can make a profit by reselling the product to the consumers who buy at high prices Thus firms resort to various marketing techniques to prevent arbitrage from taking place For example 1 Firms can charge different prices at different locations In this case in order for price discrimination to be sustained the markets Page 76 should be isolated by geography by prohibitive taxes such as tariffs or by prohibitive transportation costs such as those resulting from product spoilage while being transported from one location to another 2 Firms that provide services such as transportation companies restaurants and places of entertainment charge senior citizens lower prices than they charge younger consumers In this case for the price discrimination to be sustained the firm must demand that senior citizens present their ID cards 3 Firms can sell discount tickets to students In this case the seller will ask for a student ID card from those consumers seeking to purchase at a discount 4 Book publishers manage to charge institutions higher prices than they charge individuals by selling hardcovers to institutions and softcovers to individuals In what follows we do not analyze how the monopoly manages to segment the markets so that no arbitrage can take place between two markets with different market prices The examples given above provide some explanations of how a firm can prevent arbitrage between two markets In addition subsection 1415 demonstrates that a firm can prevent arbitrage by tying the basic product to some service for servicedemanding consumers while selling it without service to other consumers Here we merely assume that arbitrage cannot take place Consider a monopoly selling in two different markets We assume that the two markets are isolated in the sense that the monopoly can charge different prices and the consumers cannot perform arbitrage by buying in the lowprice market and selling in the highprice market We now seek to investigate how a monopoly determines the output level hence the price in each market Figure 53 illustrates the demand schedules in the two markets market I and market 2 The left side of Figure 53 illustrates the demand function and the derivedmarginalrevenue function in market 1 The middle figure illustrates the demand and marginalrevenue functions in market 2 The right side of Figure 53 illustrates the aggregate demand facing the monopoly D1 D2 and the horizontal sum of the marginalrevenue functions Σ MR The monopoly chooses the output levels sold in each market and that solve Page 77 Figure 53 Monopoly discriminating between two markets If the monopoly sells a strictly positive amount in each market then the following two firstorder conditions are satisfied Hence the discriminating monopoly equates when it sells the profitmaximizing output levels in each market The intuition behind this condition is as follows If the monopoly chooses and such that then it is clear that the monopoly should transfer one unit from market 2 to market 1 In this case the reduction in revenue in market 2 is smaller than the increase in revenue in market 1 To solve for the profitmaximizing output levels and we need to solve two equations with the two variables given in 53 Instead we provide a threestep graphical illustration for how to solve this problem First note that Figure 53 illustrates how the total production level is determined by the intersection of ΣMR with the to determine the aggregate production level Second to geometrically find the output level sold in each market draw a horizontal line from the intersection of ΣMRi MCQm to the MR1 and MR2 functions This determines the amount of output sold in each market and Third to find the price charged in each market note that consumers are always on their demand curves hence extend vertical lines from and to the corresponding demand curves to locate and Finally to find the relationship between the price charged in each market and the demand elasticities Proposition 33 and equation 53 Page 78 imply that Hence if η2 η1 or η2 η1 recalling that elasticity is a negative number Hence Proposition 51 A discriminating monopoly selling a strictly positive amount in each market will charge a higher price at the market with the less elastic demand 54 The Cartel and the Multiplant Monopoly The cartel and the multiplant monopoly are forms of organizations and contractual agreements among plants firms or countries For example if we view the oilproducing countries as plants the cartel is an organization that contracts with the countries on how much each would produce and hence on what would be the world price Other examples of cartels include the IATA International Air Transport Association which regulates airfares and bar associations which regulate attorneys The multiplant monopoly is very similar to the cartel except that all the plants are put under a single ownership Multiplant monopoly occurs when several firms in the industry merge together into a single firm horizontal merger or when a monopoly firm opens several plants producing the same product Thus unlike the cartel the multiplant monopoly has the power to decide whether to shut down some of its plants or whether to open several more A cartel generally does not shut down plants or countries for the simple legal reason that the cartel does not own the plants and no plant would join the cartel knowing that it could be shut down We assume a linear aggregate demand given by p a bQ We now define the technology of each plant We assume that there are N plants indexed by i i 1 2 N Let qi denote the output level of plant i and assume that each plant has the technology summarized by the total cost function given by Thus we assume that all plants have identical cost functions and that each plant has a fixed output independent cost of F The plants average and marginalcost functions are given by ATCiqi Fqi cqi and MCi qi 2cqi Figure 51 on page 73 illustrates this cost structure which is common to all plants 541 The cartel The cartel organizes all the N plants by directing each plant to produce a certain amount The objective is to maximize the Sum of the profits Page 79 of all the N plants Let πiqi denote the profit of plant i and let the aggregate cartel output be denoted by Q The objective of the cartel is to choose q1 q2 qN to The cartel has to solve for N quantities so after some manipulations the N firstorder conditions are given by Thus Proposition 52 The cartels profitmaximizing output produced by each plant is found by equating the marginal revenue function derived from the market demand curve evaluated at the aggregate carteloutput level to the marginalcost function of each plant Since all plants have identical cost functions we search for a symmetric equilibrium where the cartel directs each plant to produce the same output level That is Hence The total cartels output and the market price are given by Notice that when N 1 the cartels output and price coincide with the pure monopoly levels It can be easily verified that as the number of firms in the cartel increases N increases both the output level of each firm and the market price fall q and p decrease Hence the total revenue and profit of each firm must fall with an increase in the number of cartel members For this reason many professional organizations such as those of lawyers and accountants impose restrictions on new candidates who wish to practice in their profession Page 80 542 The multiplant monopoly The multiplant monopoly is very similar to the cartel except that it has the authority ownership to shut down some plants thereby saving variable and fixed costs associated with maintaining the plant Thus if we suppose that the multiplant monopoly can choose the number of plants that is N is a choice variable by the multiplant monopoly owner then the question is What is the profit maximizing number of plants operated by the multiplant monopoly The answer is very simple given that the multiplant monopoly can add or discard plants the monopoly would seek to adjust the number of plants to minimize the cost per unit of production In other words the multiplant monopoly will adjust the number of plants to minimize ATCqi for every plant in operation In order to demonstrate how the number of plants is determined we approximate the number of firms by a real continuous number rather than by an integer number Like the cartel the multiplant monopoly would equate MRQ MCiqi for every operating plant yielding output levels given in 57 equal to and in addition will adjust N so that each operating plant would operate at minimum ATCiqi given by Hence equating and solving for N yields that the profitmaximizing number of plants is Thus the multiplant monopolys profitmaximizing number of plants increases with an increase in the demand parameter a and decreases with the fixed cost parameter of each plant F 55 DurableGoods Monopolies Our analysis so far has focused on one type of goods called flow goods By flow goods we mean goods that are purchased repeatedly and that perish after usage for example food products such as apples and bananas and many plastic and paper singleuse products In contrast durable goods are bought only once in a long time and can be used for long time for example cars houses and land Clearly with the exception of land all goods eventually perish so these two concepts are relative to a certain time horizon that is relevant to consumers Coase 1972 first pointed out that a monopoly selling a durable good will behave differently from the familiar monopoly selling a perishable good analyzed earlier in this chapter Coase considered the extreme case of a person who owns all the land in the world and wants to sell it at the Page 81 largest discounted profit Clearly Coase chose to analyze land because it is definitely a good example of a durable good If land were perishable then our analysis implies that the monopoly would not sell all the land That is the monopoly would restrict output land and raise the price high enough so that not all the land would be sold Now suppose that the monopoly charges the monopoly price and sells half of its land by the end of this year Let us try to predict what will happen next year Well the monopoly still owns the remainder of the worlds land and there is no reason why the monopoly will not offer that land for sale next year However it is clear that next year if population is not growing very fast the demand for land will be lower than the demand for land this year Thus the monopoly land price next year will be lower than the monopoly price this year Given that the monopolys nextyear price will be substantially lower than the monopoly land price this year it is clear that those consumers who do not discount time too heavily would postpone buying land until next year Hence the current demand facing the monopoly falls implying that the monopoly will charge a lower price than what a monopoly selling a perishable would charge Coases discussion of durable goods monopolies was formalized in Bagnoli Salant and Swierzbinski 1989 Bulow 1982 1986 Gul Sonnenschein and Wilson 1986 and Stokey 1981 In what follows we provide two simple but rigorous analyses of durablegoods monopolies Subsection 551 demonstrates Coases conjecture in an example for a downward sloping demand curve Subsection 552 provides an example for a discrete demand in which there is a finite number of consumers each buys at most one unit of a durable good and demonstrates that Coases analysis is false under this demand structure 551 Durablegood monopoly facing a downward sloping demand Suppose there is a continuum of consumers having different valuations for the annual services of a car that are summarized by the familiar downward sloping demand curve Suppose that consumers live for two periods denoted by t t 1 2 and that a monopoly sells a durable product that lasts for two periods Thus if a consumer purchases the product she will have it for her entire life and she will not have to replace it ever again The consumers have different valuations for the product summarized by the aggregate period t 1 inverse demand function for one period of service given by p 100 Q and illustrated in Figure 54 left Figure 54 assumes that in period 1 there is a continuum of consumers each having a different valuation for purchasing Page 82 Figure 54 Durablegood monopoly the case of downward sloping demand one unit of the product Altogether they form a downward sloping demand illustrated in Figure 54 left In the following two subsubsections we compare the monopolys profit under two types of commercial transactions selling and renting To formally distinguish between selling and renting we state the following definition Definition 51 1 By selling a product to a consumer for a price of pS the firm transfers all rights of ownership for using the product and getting the product back from the consumer from the time of purchase extended indefinitely 2 By renting a product to a consumer for a price of pR the renter maintains ownership of the product but contracts with the consumer to allow the consumer to derive services from the product for a given period specified in the renting contract Thus selling means charging a single price for an indefinite period whereas renting means charging a price for using the product for a specific limited time period It should be emphasized that Definition 51 does not imply that by selling the manufacturer always transfers all rights on the product sold For example even when a product is sold rather than rented the new owner does not have the rights to produce identical or similar products if the product is under patent protection Page 83 A renting monopoly Assume that each period the monopoly rents a durable product for one period only For example a common practice of firms in several industries in particular in the car industry is to lease a car for a given time period rather than sell the car Although there could be several explanations taxes etc why such a trade benefits firms and consumers in this subsection we prove that leasing would yield a higher profit than selling Suppose that in each of the two periods the monopoly faces the demand drawn in Figure 54 left Assuming zero production cost we recall from section 51 that the monopoly would rent an amount determined by the condition MRQt 100 2Qt 0 MCQt implying that and and for t 12 Hence the lifetime sum of profits of the renting monopoly is given by πR 5000 A seller monopoly A seller monopoly knows that those consumers who purchase the durable good in t 1 will not repurchase in period t 2 That is in t 2 the monopoly will face a demand for its product that is lower than the period I demand by exactly the amount it sold in t 1 Therefore in period 2 the monopoly will have to sell at a lower price resulting from a lower demand caused by its own earlier sales Formally we define this twoperiod game as follows The payoff to the monopoly is the total revenue generated by period I and period 2 sales The strategies of the seller are the prices set in period 1 p1 and the price set in period 2 as a function of the amount purchased in period 1 The strategies of the buyers are to buy or not to buy as a function of first period price and to buy or not to buy as a function of second period price We look for a SPE for this simple game see Definition 29 on page 26 The methodology for solving this finite horizon game is to solve it backwardsto determine how the monopolist would behave in period 2 for each possible set of buyers remaining then The second period Figure 54 right shows the residual demand facing the monopoly in period 2 after it has sold units in period 1 given by or Since production was assumed to be costless in the secondperiod the monopoly sets implying that Hence the second period price and profit levels are given by and Page 84 The first period Suppose that the monopolist sells in the first period to buyers with the highest reservation prices Then the marginal buyer with a reservation price will be indifferent between purchasing in the first period gaining utility of and buying in the second period gaining utility of Thus Solving 59 for p1 yields Let us note that equation 510 can also be derived by observing that the firstperiod price should include the secondperiod price in addition to pricing the firstperiod services because buying in the first period yields services for the two periods hence the product can be resold in the second period for a price of p2 Therefore which is identical to 510 In a SPE the selling monopoly chooses a firstperiod output level that solves yielding a firstorder condition given by Denoting the solution values by a superscript S we have that and Hence Therefore Proposition 53 A monopoly selling a durable goods earns a lower profit than a renting monopoly The intuition behind Proposition 53 is that rational consumers are able to calculate that a selling durablegood monopoly would lower future Page 85 prices due to future fall in the demand resulting from having some consumers purchasing the durable product in earlier periods This calculation reduces the willingness of consumers to pay high prices in the first period the monopoly offers the product for sale In other words since the monopoly cannot commit itself not to reduce future prices the monopoly is induced to lower its firstperiod price An argument such as Proposition 53 led some economists to claim that monopolies have the incentives to produce less than an optimal level of durability eg light bulbs that burn very fast We discuss the invalidity of this argument in section 123 552 Durablegood monopoly facing a discrete demand The analysis of subsection 551 has confined itself to a demand curve with a continuum of nonatomic buyers Following Bagnoli Salant and Swierzbinski 1989 we now provide an example which demonstrates that Coases Conjecture is false when the number of consumers is finite Let us consider an economy with two consumers living only for two periods Both consumers desire car services for the two periods of their lives however the consumers differ in their willingness to pay for car services The maximum amount a consumer denoted by H is willing to pay for one period of car service is VH and the maximum amount a consumer denoted by L is willing to pay for one period of car service is VL We assume that the consumers willingness to pay per period of car service are substantially different Assumption 51 Type H consumers are willing to pay more than twice as much for a period of car service as type L consumers Formally VH 2VL 0 Figure 55 left illustrates the aggregate inverse demand function for one period of service facing the monopoly each period Because the product is durable consumers buy it once in their life either at t 1 or t 2 The utility functions for consumers type i H L that yield the demand structure illustrated in Figure 55 are given by Thus if consumer i i H L buys a car in the first period he gains a benefit of 2Vi since the car provides services for two periods and he pays whatever the monopoly charges in t 1 In contrast if the consumer Page 86 Figure 55 Durablegood monopoly the case of discrete demand waits and purchases the car in t 2 he gains only one period of utility of Vi minus the price charged in period 2 On the production side we assume that there is only one firm producing cars at zero cost Like the consumers the monopoly firm lives for two periods and maximizes the sum of profits from the sales during the two periods We denote by qt the amount produced and sold by the monopoly and by pt the period t price of a car set by the monopoly in period t t 1 2 The monopoly chooses p1 and p2 to maximize the sum of revenue from two periods worth of sales given by Note that we have implicitly assumed that buyers and the monopoly do not discount future utility and profit since assuming otherwise would not have a qualitative effect on the results A renting monopoly Suppose now that the monopoly firm does not sell cars but instead rents cars for one period only Thus each consumer who rents a car in t 1 has to return the car at the end of the first period and rent it again in the second period We denote by the rental price for one period of renting in period t Since car rentals last for one period only it is sufficient to calculate the price for each period separately Since the renting firm is a monopoly it has two options 1 setting which by 512 induces only consumer H to rent a car each period while consumer L will not rent 2 setting which induces both consumers to rent a car each period In the first case the twoperiod profit is πR 2VH and in Page 87 the second case πR 4VL However by Assumption 51 VH 2VL Hence Proposition 54 A renting monopoly would rent cars only to the highvaluation consumer by setting a rental price equal to t 1 2 and it trill earn a twoperiod profit of πR 2VH A seller monopoly Now suppose that the monopoly sells the cars to consumers We denote the selling prices by t 1 2 By Definition 51 the period 1 selling price means that the consumer pays for two periods of using the car compared with the renting price that entitles the consumer to use the car for period 1 only The second period The effect of selling in the first period on the second period demand is illustrated in Figure 55 right If consumer H purchases in period 1 only consumer L demands a car in the second period If consumer H does not purchase in the first period then the second period demand is the given rental demand curve Figure 55 left The lower part of Figure 56 illustrates the subgames associated with consumer Hs decision whether to purchase in the first period Figure 56 Twoperiod game of a durablegood monopoly facing discrete demand Figure 56 illustrates that when consumer H buys in the first period the monopoly will maximize second period profit by setting Page 88 and will earn a second period profit of π2 VL the monopoly will extract all surplus from consumer L If consumer H does not buy in period 1 then in the second period the monopoly faces the entire demand hence by Assumption 51 the monopoly charges selling only to consumer H yielding a second period profit of π2 VH The first period In the first period the monopoly sets and consumers decide whether to purchase or not Figure 56 illustrates the Sequence of moves in the two periods Since consumer L knows that the price in the last period will never fall below VL consumer L will buy in the first period at any price below 2VL Hence if the seller sets both consumers would purchase initially To simplify the game tree we report in Figure 56 the payoffs to the three players if the monopoly sets this low price Clearly the monopoly will not set because this price exceeds the two period sum of consumer Hs valuation Therefore we now check whether is the profit maximizing first period price for the seller monopoly From the second period analysis we conclude that consumer H earns a utility of zero UH 0 whether or not he buys the product in the first period Hence buying the product is an optimal response for consumer H to the first period price Thus in a SPE see Definition 210 on page 27 consumer H buys in period 1 and consumer L buys in period 2 constitute a SPE equilibrium path for this game Hence in contrast to Proposition 53 we now state our main proposition which demonstrates that Coases conjecture is false under discrete demand Proposition 55 A durablegood selling monopoly facing a discrete demand will 1 charge a first period selling price that is equal to the sum of the perperiod rental prices 2 earn a higher profit than the renting monopoly that is πS 2VH VL 2VH πR Thus in the case of discrete demand a selling monopoly can extract a higher surplus from consumers than the renting monopoly Coase conjectured that the ability of a durablegood monopoly to extract consumer surplus is reduced when the monopoly is forced to sell rather than rent Here we demonstrated the opposite case where selling enables the monopoly to price discriminate among different consumers by Page 89 setting prices which would induce different consumers to purchase at different time periods 56 Appendix The Legal Approach to Monopoly and Price Discrimination 561 Antitrust law and the monopoly Section 2 of the Sherman Act of 1890 states that Every person who shall monopolize or attempt to monopolize or combine and conspire with any other person or persons to monopolize any part of the trade or commerce among several States or with foreign nations shall be deemed guilty of a felony At first glance it seems that section 2 makes it clear that a monopoly is illegal but a closer look reveals that the act does not provide the court with any guidelines that define what degree of market power or market concentration constitutes a monopoly Therefore in practice courts tend to focus on abuses of monopoly power in a concentrated market and on the intent of the monopoly to keep its position monopoly status alone is not illegal Anticompetitive activities such as predatory pricing have to be established to turn a monopoly into an illegal practice To establish illegal activities the court first defines the product and the geographic market Second the court considers the market share of the accused firm Third the court considers the ease of entry availability of secondhand and new substitutes and whether the accused has the ability to raise prices Defining the product is basically deciding which products should be considered as close substitutes Deigning the geographic market should consider the magnitude of transportation costs which in many cases are insignificant thereby leading the court to define the entire nation as the geographic market When these tests are unclear the court resorts to a hypothetical question In a particular geographic market can the accused firm raise the price without attracting competition If the answer is positive then the market is well defined During the years courts have added a refusal to deal when a manufacturer refuses to sell to dealers for the purpose of establishing a monopoly power on all distribution channels as an abuse of monopoly power 562 Antitrust law and cartels Cartels may involve price fixing output controls bid rigging allocation of consumers allocation of sales by product or territory establishment Page 90 of trade practices or common sales agencies Weiss 1987 Cartels have existed as guilds in the Europe of the Middle Ages and were common in most European countries throughout the nineteen century and the first third of the present century The Sherman Act of 1890 made cartel illegal Exceptions were made during the Great Depression and for some special quasipublic industries such as agriculture coal civil aviation and oil refining Section 1 of the Sherman Act 1890 states that Every contract combination in the form of a trust or otherwise or conspiracy in restraint of trade or commerce among the several States or with foreign nations is declared to be illegal Clearly the most severe and most common cartel contract is a pricefixing contract Firms that are found guilty of price fixing are subject to trebledamage penalties Recently several authors raised the question of whether trebledamage penalty would result in marketprice reduction or market price increase Salant 1987 showed that trebledamage penalty can increase the market price above the price that would be charged by a cartel without the enforcement of this antitrust law Earlier court cases interpreted section I to mean that every contract constituted a restraint of trade thereby leading courts to rule on a per se basis defined in subsection 122 on page 6 of this book That is every price fixing was illegal In some later cases courts considered some pricefixing arrangements under the rule of reason However courts began learning that any judgment under the rule of reason involves tremendous administrative costs since it is not clear what a reasonable price is and it is hard to measure marginalcost functions to determine whether the price is fixed with a high markup It was also clear that prices should often fluctuate with cost variations something that may not occur in the presence of price fixing Hence courts began judging price fixing under the per se rule The logic was that if pricefixing agreements do not have an effect on prices then these agreements would not be formed Thus pricefixing agreements should be illegal per se The per se rule was also applied to other forms of contracts such as market allocations Finally one advantage of the per se rule is that it warns the firms in advance about the consequences generally treble damages associated with pricefixing agreements whereas the rule of reason may leave some doubts whether with a good defense a cartel can survive section I in a lawsuit To summarize we can say that the major effect of section 1 of the Sherman Act is rather noticeable The act indeed eliminated major cartels from American markets Most noticeable cartels nowadays for Page 91 example OPEC and IATA are international and cannot be challenged for rather visible price fixing agreements 563 Antitrust law and price discrimination Section 2 of the Clayton Act of 1914 amended by the RobinsonPatman Act of 1936 states that It shall be unlawful for any person engaged in commerce in the course of such commerce either directly or indirectly to discriminate in price between different purchasers of commodities of like grade and quality where the effect of such discrimination may be substantially to lessen competition or tend to create a monopoly in any line of commerce or to injure destroy or prevent competition with any person who either grants or knowingly receives the benefit of such discrimination or with the consumers of either of them Provided That nothing herein contained shall prevent differentials who make only due allowance for differences in the cost of manufacture sale or delivery Thus section 2 explicitly states that price discrimination should not be considered illegal a unless price discrimination substantially decreases competition and b if price differences result from differences in production or delivery costs Thus the coupons appearing in the Sunday newspapers offering a price reduction upon the presentation of a piece of paper is a good example of price discrimination between people with a high value on time and a low value on time however there is nothing illegal in using coupons for providing discounts The RobinsonPatman Act of 1936 came during the Great Depression and was intended to strike against large chain grocery stores that engaged in local price cutting to deter competition Note that at that time the legislators were not concerned whether price discrimination and price cutting are efficient In fact Varian 1989 shows conditions under which the act of price discrimination is welfare improving compared with a uniform price mechanism Also note that the GATT General Agreement on Tariff and Trade enacted a rule similar to the one enacted in RobinsonPatman stating that dumping selling below cost in a foreign country is illegal However it has never been theoretically established that dumping reduces welfare and it is possible to demonstrate that dumping can actually improve social welfare Altogether it is not clear whether price discrimination has anything to do with anticompetitive behavior and in fact price discrimination can actually be procompetitive Bork 1978 warns against possible Page 92 damages inflicted by this act by conjecturing that there may be hundreds of thousands of pricing decisions every year that are altered through fear of the RobinsonPatman Act meaning that hundreds of thousands of quantity discounts and promotional discounts are foregone at the expense of having consumers paying higher prices However during the years following this act the FTC rarely enforced this law thereby making price differences more observable 57 Exercises 1 Consider a monopoly selling at a single market where the demand is given by Suppose that the cost function of this monopoly is given by c 0 a Calculate the demand elasticity and using Proposition 33 write down the marginalrevenue function as a function of price b Using your above calculation find the price charged by the monopoly as a function of c What happens to the monopolys price when increases Interpret your result d What happens to the monopolys price as Explain e Calculate the totalrevenue function TRQ and the marginalrevenue function MRQ f What is the monopolys profitmaximizing output 2 Consider the market for the GJeans the latest fashion among people in their late thirties G Jeans are sold by a single firm that carries the patent for the design On the demand side there are nH 0 highincome consumers who are willing to pay a maximum amount of VH for a pair of C Jeans and nL 0 lowincome consumers who are willing to pay a maximum amount of VL for a pair of GJeans Assume that VH VL 0 and that each consumer buys only one pair of jeans Suppose that the GJeans monopoly cannot price discriminate and is therefore constrained to set a uniform market price a Draw the market aggregatedemand curve facing the monopoly b Find the profitmaximizing price set by GJeans considering all possible parameter values of nH nL VH and VL Assume that production is costless 3 Suppose that a monopoly can price discriminate between two markets market 1 where the demand curve is given by and market 2 where the demand curve is given by q2 4 p2 Suppose that once the product is sold it cannot be resold in the other market That is assume that arbitrage is impossible say due to strict custom inspections on the border between the two markets Assume that the monopoly produces each unit at a cost of c 1 Page 93 a Calculate the profitmaximizing output level that the monopoly sells in each market Calculate the price charged in each market b Calculate the monopolys profit level c Suppose that markets 1 and 2 are now open and all consumers are free to trade and to transfer the good costlessly between the markets Thus the monopoly can no longer price discriminate and has to charge a uniform price denoted by p p p1 p2 Find the profit maximizing value of p 4 A discriminating monopoly sells in two markets Assume that no arbitrage is possible The demand curve in market 1 is given by p1 100 q12 The demand curve in market 2 is given by p2 100 q2 We denote the monopolys aggregate production by Q where The monopolys cost function depends on total production and is given by TCQ Q2 Answer the following questions a Formulate the monopolys profit function as a function of q1 and q2 b Calculate the monopolys profitmaximizing quantity sold in market 1 and market 2 c Calculate the profit level of the discriminating monopoly d Suppose now that a new management assumed control of this firm The young CEO decides to decompose the monopoly plant into two plants where plant 1 sells in market 1 only and plant 2 sells in market 2 only Calculate the profitmaximizing output level sold by each plant e Calculate the sum of profits of the two plants f Conclude whether this plant decomposition increases or decreases profit Explain your answer by investigating whether the above technology exhibits increasing or decreasing returns to scale Consult Definition 32 on page 45 5 The demand elasticity in market I is measured to be η1 2 The demand elasticity in market 2 is measured to be η2 4 Suppose that a monopoly that can price discriminate between the markets sets the price p1 in market 1 and p2 in market 2 Prove whether the following statement is right or wrong The price in market 1 p1 will be 150 the price in market 2 p2 ie 50 higher 6 In a twoperiod lived economy one consumer wishes to buy a TV set in period 1 The consumer lives for two periods and is willing to pay a maximum price of 100 per period of TV usage In period 2 two consumers who live in period 2 only are born Each of the newly born consumers is willing to pay a maximum of fifty dollars for using a TV in period 2 Suppose that in this market there is only one firm producing TV sets that TV sets are durable and that production is costless Page 94 a Calculate the prices the monopoly charges for TV sets in periods 1 and 2 b Answer the previous question assuming that in the first period a consumer who lives two periods is willing to pay no more than twenty dollars per period for TV usage 7 A monopoly is facing a downward sloping linear demand curve given by p a Q The monopolys unit production cost is given by c 0 Now suppose that the government imposes a specific tax of t dollars per unit on each unit of output sold to consumers a Show that this tax imposition would raise the price paid by consumers by less than t Hint One way to find the monopolys profitmaximizing output level is to solve the equation MRQ c t and then to solve for consumer and producer prices b Would your answer change if the market demand curve has a constant elasticity and is given by 58 References Bagnoli M S Salant and J Swierzbinski 1989 DurableGoods Monopoly with Discrete Demand Journal of Political Economy 97 14591478 Bork R 1978 The Antitrust Paradox New York Basic Books Bulow J 1982 Durable Goods Monopolists Journal of Political Economy 15 31432 Bulow J 1986 An Economic Theory of Planned Obsolescence Quarterly Journal of Economics 51 729748 Coase R 1972 Durable Goods Monopolists Journal of Law and Economics 15 143150 Gellhorn E 1986 Antitrust Law and Economics in a Nutshell St Paul Minn West Publishing Gul F H Sonnenschein and R Wilson 1986 Foundations of Dynamic Monopoly and the Coase Conjecture Journal of Economic Theory 39 155190 Krueger A 1974 The Political Economy of the RentSeeking Society American Economic Review 64 291303 Posner R 1975 The Social Costs of Monopoly and Regulation Journal of Political Economy 83 807827 Salant S 1987 Treble Damage Awards in Private Lawsuits for Price Fixing Journal of Political Economy 95 13261336 Stokey N 1981 Rational Expectations and Durable Goods Pricing Bell Journal of Economics 12 112128 Tullock G 1967 The Welfare Costs of Tariffs Monopolies and Theft Western Economic Journal 5 224232 Page 95 Varian H 1989 Price Discrimination In Handbook of Industrial Organization edited by R Schmalensee and R Willig Amsterdam NorthHolland Weiss L 1987 Cartel In The New Palgrave Dictionary of Economics edited by J Eatwell M Milgate and P Newman New York The Stockton Press Page 97 Chapter 6 Markets for Homogeneous Products Only theory can separate the competitive from the anticompetitive Robert Bork The Antitrust Paradox In this chapter we analyze the behavior of firms and consumer welfare under several oligopolistic market structures The main assumption in this chapter is that the products are homogeneous meaning that consumers cannot differentiate among brands or distinguish among the producers when purchasing a specific product More precisely consumers cannot or just do not bother to read the label with the producers name on the product they buy For example nonbrandname products sold in most supermarketsbulk fruit vegetables containers of grainare generally purchased without having consumers learning the producers name In what follows we assume that consumers are always price takers henceforth competitive and have a welldefined aggregatedemand function However firms behave according to the assumed market structures analyzed below Our oligopoly analysis starts with section 61 Cournot which assumes that firms set their output levels simultaneously believing that the output levels of their rival firms remain unchanged Historically as we discuss below Cournot was the first to provide this modern treatment of oligopoly equilibrium Section 62 Sequential Moves modifies the static Cournot setup by assuming that firms move in sequence and analyzes whether a firm benefits by setting its output level before any other one does Following Bertrands criticism of the use of quantity produced as the actions chosen by firms section 63 Bertrand analyzes Page 98 a market structure where firms set their prices by assuming that the prices of their rival firms remain unchanged We then discuss how the extreme result of price games leading to competitive prices obtained under the Bertrand competition can be mitigated by introducing capacity constraints Section 64 Cournot Versus Bertrand analyzes the relationship between the Cournot and the Bertrand market structures Section 65 SelfEnforcing Collusion analyzes the conditions under which firms can maintain higher prices and lower output levels compared with the Cournot levels assuming that the firms interact infinitely many times Section 66 International Trade analyzes international markets in homogeneous products 61 Cournot Market Structure Noncooperative oligopoly theory started with Antoine Augustin Cournots book Researches into the Mathematical Principles of the Theory of Wealth published in France in 1838 In that book Cournot proposed an oligopolyanalysis method that we today view as identical to finding a Nash equilibrium in a game where firms use their production levels as strategies Cournot earned his doctorate in science in 1821 with a main thesis in mechanics and astronomy Cournots writings extended beyond economics to mathematics and philosophy of science and philosophy of history see Shubik 1987 Cournot was central to the founding of modern mathematical economics For the case of monopoly the familiar condition where marginal revenue equals marginal cost come directly from Cournots work Shubik 1987 In chapter 7 of his book Cournot employs the inversedemand function to construct a system of firms marginalrevenue functions which could be then solved for what we will call the Cournot output levels Then he introduced firms cost functions and the system of first order conditions to be solved Cournot did not consider the possibility that firms with sufficiently high cost may not be producing in this equilibrium In what follows we develop the Cournot oligopoly model where firms sell identical products In this model firms are not price takers Instead each firm is fully aware that changing its output level will affect the market price 611 Twoseller game Let us consider a twofirm industry summarized by the cost function of each firm i producing qi units given by Page 99 and the marketdemand function given by In contrast to chapter 4 where we solved for a competitive equilibrium for this industry here we solve for a Cournot oligopoly equilibrium We first have to define a twofirm game that corresponds to a definition of a game given in Definition 21 Let each firms action be defined as choosing its production level and assume that both firms choose their actions simultaneously Thus each firm i chooses i 1 2 Also let the payoff function of each firm i be its profit function defined by πiq1 q2 pq1 q2qi TCiqi Now the game is properly defined since the players their action sets and their payoff functions are explicitly defined All that is left to do now is to define the equilibrium concept Definition 61 The triplet is a CournotNash equilibrium if That is according to Definition 61 a Cournot equilibrium is a list of output levels produced by each firm and the resulting market price so that no firm could increase its profit by changing its output level given that other firms produced the Cournot output levels Thus Cournot equilibrium output levels constitute a Nash equilibrium in a game where firms choose output levels Now that the equilibrium concept is well defined we are left to calculate the Cournot equilibrium for this industry Firm 1s profitmaximization problem yields the firstorder condition given by which yields the familiar profitmaximizing condition in which each firm firm 1 in this equation sets its marginal revenue MRq1 a 2bq1 bq2 equal to marginal cost c1 The secondorder condition guaranteeing a global maximum is satisfied since for every q1 and q2 Solving for q1 as a function of q2 yields the bestresponse Page 100 function also commonly known as reaction function of firm 1 which we denote by R1 q2 Hence Similarly we can guess that firm 2s bestresponse function is given by The bestresponse functions of the two firms are drawn in Figure 61 in the q1q2 space Figure 61 Cournot bestresponse functions the case for c2 c1 The two bestresponse functions are downward sloping implying that for each firm if the rivals output level increases the firm would lower its output level The intuition is that if one firm raises its output level the price would drop and hence in order to maintain a high price the other firm would find it profitable to decrease its output level A perhaps more intuitive explanation for why a firms bestresponse function is downward sloping is that an increase in a rivals output shifts the residual demand facing a firm inward Hence when a firm faces a lower demand it would produce a smaller amount Now the Cournot equilibrium output levels can be calculated by solving the two bestresponse functions 63 and 64 which correspond to the intersection of the curves illustrated in Figure 61 Thus Page 101 Hence the aggregate industryoutput level is and the Cournot equilibrium price is It is easy to confirm from 65 that the output of the highcost firm is lower than the output level of the lowcost firm That is implies that Altogether the Cournot profit payoff level of firm i as a function of the unit costs for firms i and j is given by We conclude this section with some comparative static analysis Suppose that firm 1 invents a new production process that reduces its unit production cost from c1 to where The type of RD leading to cost reduction is called process innovation to which we will return in Chapter 9 Equation 65 implies that increases while decreases This is also shown in Figure 61 where a decrease in c1 shifts R1 q2 to the right thereby increasing the equilibrium while decreasing Also 66 implies that a decrease in c1 or c2 would decrease the equilibrium price pc and 67 implies that a decrease in c1 would increase the profit of firm 1 while lowering the profit of firm 2 612 Nseller game Suppose now the industry consists of N firms We analyze two types of such industries a N identical firms all having the same cost function or b heterogeneous firms where some firms have cost functions different from others Since solving the general case of firms with different cost functions would require solving N firstorder conditions intersecting N bestresponse functions we first solve the model by assuming that all firms have identical technologies That is ci c for every i 1 2 N In the appendix section 67 we introducea procedure that makes solving the heterogeneousfirms case easy Since all firms have the same cost structure the first step would be to pick up one firm and calculate its output level as a function of the output levels of all other firms In other words we would like to calculate the bestresponse function of a representative firm With no loss of generality we derive the bestresponse function of firm 1 Thus Page 102 firm 1 chooses q1 to The firstorder condition is given by Hence the bestresponse function of firm 1 as a function of the output levels of firms q2 q3 qN is given by In the general case where firms have different cost functions we would have to derive the best response function for each of the N firms However since all firms are identical we can guess that in a Cournot equilibrium the firms would produce the same output levels we guess and later verify that Thus we denote the common output level by q where q qi for every i Note that a common mistake among students is to substitute q for qi before the best response functions are derived This procedure is obviously leading to the wrong solution since it implies that each firm controls the output level of all firms Therefore here we substitute the common q only into the already derived bestresponse functions The use of symmetry here is purely technical and is done to facilitate solving N equations with N unknowns From 68 we have it that Hence The equilibrium price and the profit level of each firm are given by Varying the number of firms We now ask how would the Cournot price quantity produced and profit levels change when we change the number of firms in the industry First note that substituting N 1 into 69 and 610 yields the monopoly solution described in section 51 Second substituting N 2 yields the duopoly solution described in 65 66 and 67 Page 103 Now we let the number of firms grow with no bounds Then we have it that That is in a Cournot equilibrium as the number of firms grows indefinitely the output level of each firm approves zero whereas the industrys aggregate output level approaches the competitive output level given in Proposition 41 Also Hence the Cournot equilibrium price approaches the competitive price that equals the unit production cost of a firm see Proposition 41 These resets often cause some confusion among students leading them to believe that competitive behavior occurs only when there are many or infinitely many firms However as we pointed out in chapter 4 we can assume a competitive market structure for any given number of firms and even solve for a competitive equilibrium for the case where N 1 What equations 611 and 612 say is that the Cournot market structure yields approximately the same price and industry output as the competitive market structure when the number of firms is large 613 Cournot equilibrium and welfare Since our analysis starts with given demand functions rather than the consumers utility functions we cannot measure the social welfare by calculating consumers equilibriumutility levels Instead we approximate social welfare by adding consumers surplus and firms profits see subsection 323 on page 52 for a justification of this procedure of welfare approximation Note that profit shoed be part of the economys welfare because the firms are owned by the consumers who collect the profits via firms distributions of dividend Substituting the Cournot equilibrium price 610 into 33 on page 52 we obtain the consumers surplus as a function of the number of firms N Hence Clearly meaning that consumers surplus rises with the entry of more firms due to the reduction in price and the increase in the quantity consumed We define social welfare as the sum of consumers surplus plus the industry aggregate profit see section 43 on page 68 for a definition Thus if we recall 610 Page 104 Also note that Hence although the industry profit declines with an increase in the number of firms the increase in consumers surplus dominates the reduction in the industry profit Thus in this economy free entry is welfare improving 62 Sequential Moves In the previous section we analyzed industries where firms strategically choose their output levels All those games were static in the sense that players simultaneously choose their quantity produced In this section we assume that the firms move in sequence For example in a twofirm sequential moves game firm 1 will choose its output level before firm 2 does Then firm 2 after observing the output level chosen by firm 1 will choose its output level and only then will output be sold and profits collected by the two firms This type of market structure is often referred to as Leader Follower on the basis of yon Stackelbergs work 1934 see Konow 1994 for yon Stackelbergs biography This type of behavior defines an extensive form game studied in section 22 In this section we do not raise the important question of what determines the order of moves that is why one firm gets to choose its output level before another We return to this question in chapter 8 where we distinguish among established firms called incumbent firms and potential entrants Here we assume that the order of moves is given and we develop the tools for solving an industry equilibrium under a predetermined order of moves We analyze a twostage game where firm 1 the leader chooses the quantity produced in the first stage The quantity chosen in the first stage is irreversible and cannot be adjusted in the second stage In the second stage only firm 2 the follower chooses how much to produce after observing the output level chosen by firm 1 in the first stage Here the game ends after the second stage and each firm collects its profit Our main questions are a Is there any advantage for moving in the first stage rather than the second and b How would the equilibrium market price and production levels compare to the static Cournot equilibrium price and output levels Following Definition 29 on page 26 this game has a continuum of subgames indexed by the output level chosen by firm 1 in the first stage A finitehorizon dynamic game is generally solved backwards We look for a subgame perfect equilibrium Definition 210 on page 27 for this game Hence we first analyze the players firm 2 in our case action in Page 105 the last period assuming that the actions played in previous period are given Then we go one period backwards and analyze firm ls action given the strategy see Definition 28 on page 24 of how firm 2 chooses its output level based on the firstperiod action To simplify the exposition let all firms have identical unit cost c1 c2 c The secondperiod subgames In the second period only firm 2 moves and chooses q2 to maximize its profit taking firm ls quantity produced q1 as given As you probably noticed we have already solved this problem before since the secondperiod problem of firm 2 is identical to the problem firm 2 solves in a Cournot market structure This maximization results in the bestresponse function of firm 2 given in 64 Hence Note that the function R2 q1 constitutes firm 2s strategy for this game since it specifies its action for every possible action chosen by firm 1 The firstperiod game In period 1 firm 1 calculates R2q1 in the same way as firm 2 Thus firm 1 is able to calculate how firm 2 will best reply to its choice of output level Knowing that firm 1 chooses to We leave it to the reader to derive the first and secondorder conditions Thus the quantity produced by the leader is Hence under the sequentialmoves market structure the leader produces a higher level of output than the Cournot market structure Substituting 615 into R2q1 yields the followers equilibrium output level implying that the followers output level falls compared with the Cournot output level Thus the leaders gain in output expansion comes partly from the reduction in the followers output level The equilibrium price and aggregate output levels are given by Therefore Page 106 Proposition 61 A sequentialmoves quantity game yields a higher aggregate industryoutput level and a lower market price than the static Cournot market structure Thus the equilibrium market outcome under a sequentialmoves game is more competitive than the Cournot equilibrium outcome in the sense that this outcome is somewhere in between the competitive equilibrium outcome derived in chapter 4 and the Cournot outcome derived in section 61 The intuition behind Proposition 61 is as follow Under the Cournot market structure firm I perceives the output produced by firm 2 as given However under sequentialmoves market structure firm I knows firm 2s bestresponse function and therefore calculates that firm 2 will reduce its output level in response to its increase in output level Hence when firm 1 expands output it expects the price to fall faster under Cournot than under sequentialmoves market structure Therefore in order maintain a high price firm I will produce more under the sequential game than it will under Cournot Now 615 and 616 demonstrate that the increase in aggregate output stems from the fact that the follower does not find it profitable to cut its output level by the same amount as the increase in the leaders output level This happens because the reaction functions are sloped relatively flat slope is negative but exceeds1 implying that a firm reduces its output level by less than the increase in the output level of the rival firm We now compare firms profit levels under sequential moves to the Cournot profit levels We leave it to the reader to verify that the leaders profit increases while the followers declines That is where and are given in 67 Note that we could have concluded even without going into the precise calculations that the leaders profit under the sequentialgame equilibrium will be higher than under the Cournot How It is very simple Since firm 2 reacts in a Nash fashion firm 1 could just choose to produce the Cournot output level In this case firm I would earn exactly the Cournot profit However since in the sequential game firm I chooses to produce a different output level it must be increasing its profit compared with the Cournot profit level The kind of reasoning we just described is called a revealed profitability argument and the reader is urged to learn to use this kind of reasoning whenever possible because performing calculations to investigate economic effects does not generate an intuitive explanation for these effects In contrast logical deduction often provides the necessary intuition for understanding economic phenomena Page 107 Finally we can logically deduce how industry profit under sequential moves compare with industry profit under Cournot Equations 617 show that the market price under sequential moves is lower than it is under Cournot Since the Cournot market price is lower than the monopolys price and since monopoly makes the highest possible profit it is clear that industry profit must drop when we further reduce the price below the monopolys price Hence whenever c1 c2 industry profit must be lower under sequential moves In a more general environment this argument may not holds when the industry profit is not a concave function of p 63 Bertrand Market Structure In a Cournot market structure firms were assumed to choose their output levels where the market price adjusted to clear the market and was found by substituting the quantity produced into consumers demand function In contrast in a Bertrand market structure firms set prices rather than output levels The attractive feature of the Bertrand setup compared with the Cournot market structure stems from the fact that firms are able to change prices faster and at less cost than to set quantities because changing quantities will require an adjustment of inventories which may necessitate a change in firms capacity to produce Thus in the short run quantity changes may not be feasible or may be too costly to the seller However changing prices is a relatively lowcost action that may require only a change in the labels displayed on the shelves in the store Let us turn to the Bertrand market structure In 1883 Joseph Bertrand published a review of Cournots book 1838 harshly critical of Cournots modeling It seems however that Bertrand was dissatisfied with the general modeling of oligopoly rather than with the specific model derived by Cournot Today most economists believe that quantity and price oligopoly games are both needed to understand a variety of markets That is for some markets an assumption that firms set quantities may yield the observed market price and quantity produced whereas for others a pricesetting game may yield the observed market outcomes Our job as economists would then be to decide which market structure yields a better approximation of the observed price and quantity sold in each specific market We now analyze the twofirm industry defined in 61 and 62 and look for a Nash equilibrium see Definition 24 in a game where the two firms use their prices as their actions First note that so far our analysis has concentrated on a single market price determined by our assumption that consumers are always on their demand curve However in a Bertrand game we have to consider outcomes where each firm Page 108 sets a different price for its product Therefore we now make two explicit assumptions about consumers behavior under all possible prices announced by both firms 1 Consumers always purchase from the cheapest seller 2 If two sellers charge the same price half of the consumers purchase from firm 1 and the other half purchase from firm 2 Formally we modify the demand given in 62 to capture the quantity demand faced by each firm i i 1 2 Therefore we assume that Equation 619 is the quantity demand facing firm i at any given p1 and p2 and incorporates what is commonly called a rationing rule which tells us how the market demand is divided between two firms selling a homogeneous product Thus if firm i charges a higher price than firm j then no consumer would purchase the product from firm i In contrast if pi pj then all the consumers will purchase only from firm i and none will purchase from firm j In this case the quantity demanded from firm i is calculated directly from 62 Finally if both firms charge the same prices then the quantity demand determined in 62 is equally split between the two firms Definition 62 The quadruple is a BertrandNash equilibrium if 1 given maximizes maxp1 2 given maximizes maxp2 3 q1 and q2 are determined in 619 Definition 62 states that in a BertrandNash equilibrium no firm can increase its profit by unilaterally changing its price In the next two subsections we apply Definition 62 to two types of markets the first where firms do not have capacity constraints and can produce any amount they wish under the assumed cost structure and the second where we assume that firms capacities are limited and therefore in the short run they are unable to expand production Page 109 631 Solving for Bertrand equilibrium Before we characterize the Bertrand equilibria it is important to understand the discontinuity feature of this game In the Cournot game the payoff profit functions are continuous with respect to the strategic variables quantities in the Bertrand price game by contrast equation 619 exhibits a discontinuity of the payoff functions at all the outcomes where p1 p2 That is if one firm sells at a price that is one cent higher than the other firm it would have a zero market share However a two cent price reduction by this firm would give this firm a one 100 percent market share The action of a firm to slightly reduce the price below that of its competitor is called undercutting Since undercutting involves setting a price slightly lower than the competitors we need to examine the types of currencies used in order to determine the smallest possible undercutting actions available to firms Therefore we make the following definition Definition 63 Let be the smallest possible monetary denomination smallest legal tender The medium of exchange money is said to be continuous if and discrete if Examples of discrete smallest legal tenders are in China Fen in Finland Penniä in Israel Agorot and in the US cent The following proposition characterizes Bertrand equilibria Proposition 62 1 If the medium of exchange is continuous and if the firms have the same cost structure then a Bertrand equilibrium is and 2 Let the medium of exchange be discrete and assume that c2 is denominated in the medium of exchange That is where is an integer Also let be sufficiently small that is satisfying Then if the unique Bertrand equilibrium is p2 c2 and Thus if firms have equal unit costs the Bertrand equilibrium price and aggregate output are the same as for the competitive equilibrium In other words undercutting reduces the prices to marginal cost In cases where firm 1 has a lower unit cost than firm 2 firm 1 undercuts firm 2 by charging the highest possible price that is lower than c2 which is given by Page 110 Proof Part 1 In equilibrium each firm must make nonnegative profit Hence i 1 2 We first establish that in a Bertrand equilibrium both firms charge the same prices By way of contradiction suppose that Then by 619 firm I makes zero profit However since the medium of exchange is continuous firm I can increase its profit by reducing its price to and grab the entire market thereby making strictly positive profit a contradiction By way of contradiction suppose that Then since the medium of exchange in continuous firm 2 can raise its price slightly while still maintaining a lower price than firm 1 Hence firm 2 will deviate a contradiction Now that we have established that by way of contradiction assume that Clearly this cannot constitute a Nash equilibrium in prices since firm 1 say would have an incentive unilaterally to reduce its price to where can be as small as one wants thereby grabbing the entire market For sufficiently small this deviation is profitable for firm 1 Part 2 To briefly sketch the proof of part 2 observe that firm 2 makes a zero profit and cannot increase its profit by unilaterally raising its price above Hence firm 2 does not deviate Now for firm 1 to be able to sell a positive amount it must set If then 619 implies that the firms split the market by selling each In this case the profit of firm 1 is However if firm I undercuts by the smallest legal tender then it becomes the sole seller and sells In this case Comparing 620 with 621 yields the condition stated in part 2 632 Bertrand under capacity constraints The previous section demonstrated that when the firms have the same cost structure price competition reduces prices to unit costs thereby making firms earn zero profits Economists often feel uncomfortable with this result especially since it makes the number of firms in the industry irrelevant in the sense that under symmetric Bertrand competition price drops to unit cost even when there are only two firms Now if most industries are indeed engaged in a Bertrand competition Page 111 as described in this section then we should observe unitcost prices for those industries with two or more firms If this case is realistic then the antitrust authority should not have to worry about industries concentration levels and should devote all its effort to fighting monopolies Clearly we rarely observe intense price competition among industries with a small number of firms and therefore the antitrust authority challenges mergers of firms that lead to highly concentrated industries see Section 86 One way to overcome this problem is to follow Edgeworth 1925 and to assume that in the short run firms are constrained by given capacity that limits their production levels The Irish economist Francis Ysidro Edgeworth who made enormous contributions to economic theory and other disciplines identified some discontinuity properties of the firms profit functions when firms produce under increasing marginal cost decreasing returns to scale technologies In Edgeworths words Edgeworth 1925 118 In the last case there will be an intermediate tract through which the index of value will oscillate or rather vibrate irregularly for an indefinite length of time There will never be reached that determinate position of equilibrium which is characteristic of perfect competition We demonstrate Edgeworths argument by assuming an extreme version of increasing marginal cost which is letting the cost of expanding production beyond a certain output level which we call capacity be infinite Figure 62 illustrates a marketdemand curve composed of four consumers each buying at most one unit Figure 62 assumes that consumer I is willing to pay a maximum of 3 for one unit consumer 2 a maximum of 2 consumer 3 a maximum of 1 and consumer 4 will not pay at all Such prices are commonly termed as consumers reservation prices Suppose now that there are two firms and that each is capable of producing at zero cost c1 c2 0 Then Proposition 62 proved in the previous subsection shows that if firms are not subject to capacity constraints then Bertrand competition would lead to prices of zero To demonstrate Edgeworths argument suppose now that in the short run each firm is limited to producing at most two units Then it is easy to show that the prices p1 p2 0 no longer constitute a Nash equilibrium To see this observe that firm 1 can increase its profit from π1 0 to π1 3 by increasing its price to p1 3 and selling its unit to the consumer with the highest reservation price In this outcome firm 1 sells one unit to the consumer with a reservation price of 3 Page 112 whereas firm 2 sells a unit to one of the other consumers for the price of p2 0 Since one firm would always want to deviate from the unit cost pricing we conclude that the Bertrand equilibrium prices under no capacity constraints need not be Nash equilibrium prices under capacity constraints Figure 62 Edgeworth Cycles Bertrand competition under capacity constraints We are left to show that in the present example there does not exist a Nash equilibrium in prices This result is sometimes referred to as Edgeworth Cycles since under any pair of firms prices one firm would always find it profitable to deviate To see this let us look at the outcome p1 3 and p2 0 Clearly firm 2 would deviate and undercut firm 1 by setting where is a small number In this case firm 1 sells nothing whereas firm 2 sells its unit to the consumer with the highest reservation price and earns a profit of Clearly firms continue undercutting each others prices and a Nash equilibrium in prices is never reached Hence we showed that marginalcost pricing is not an equilibrium under capacity constraint and that firms will keep changing prices without reaching any Nash equilibrium in prices Finally it should be pointed out that introducing capacity constraints on the firms is not the only way to generate abovemarginalcost equilibrium prices Abovemarginalcost pricing can be an equilibrium outcome a when products are differentiated see next chapter b when demand randomly fluctuates and c when firms are engaged in an infinitely pricing repeated game 64 Cournot versus Bertrand In sections 61 and 63 we analyzed the same industry where in the Cournotmarketstructure firms use quantity produced as actions whereas Page 113 in the Bertrandmarketstructure firms use prices as actions The analyses of these sections show that in general the two types of market structures yield different market outcomes prices and quantity produced Thus when we change the firms actions from choosing quantities to choosing prices the Nash equilibrium yields a completely different outcome because under Cournot firms make positive profit since the resulting market price exceeds unit cost whereas under Bertrand prices drop to unit cost Moreover in a Bertrand game only the lowcost firm produces which is generally not the case for the Cournot game Therefore we can state that in a oneshot static game there is no correspondence between the Cournot solution and the Bertrand solution However Kreps and Scheinkman 1983 constructed a particular environment a particular two period dynamic game where in the first period firms choose quantity produced accumulate inventories and in the second period the quantities are fixed cannot be changed and firms choose prices They showed that the quantities chosen by firms in the first period and the price chosen in the second period are exactly the Cournot outcome given in 65 and 66 That is they show that for some market games where two firms choose how much to produce in period 1 and then set prices in period 2 a subgame perfect equilibrium see Definition 210 on page 27 yields the exact quantity produced and price as those in a oneshot Cournotmarketstructure game where firms choose only how much to produce We will not bring a complete proof of their proposition however we illustrate the idea in our simple twofirm industry for the case where p 10 Q and both firms have a unit cost of c 1 As we discussed earlier the easiest way of solving for a subgame perfect equilibrium for a dynamic finite game is to solve it backwards Therefore we begin with the second period and ask what prices will be chosen by firms in a Nashequilibrium oneshot price game where the quantity produced is taken as given by firstperiod choices Then we analyze the first period looking for a subgame perfect equilibrium in firstperiod production levels where firms can calculate and take into account the secondperiod equilibrium market prices which depend on firstperiod production levels The secondperiod subgame Assume that for some reason the firms choose to produce the Cournot capacity levels Hence total industry output is Qc 6 We now show that in a Nash equilibrium for the second period subgame both firms will choose to set prices that clear the market under the Cournot outcome That is each firm will set pi 4 pc Figure 63 Page 114 illustrates the Cournot outcome Figure 63 Residual demand when firms have fixed inventories Note that in the second period firms are free to choose any price they wish so that the Nash equilibrium prices may differ from pc 4 To demonstrate that this is not the case we now show that given p2 4 firm 1 will not debate and will also choose p1 4 First note that firm 1 will not lower its price below p1 4 because a price reduction will not be followed by an increase in sales the capacity is limited to q1 3 The lowering the price will only lower its revenue Second we must show that firm 1 cannot increase its profit by raising its price and selling less than The right side of Figure 63 exhibits the residual demand facing firm 1 when it raises its price above Residual demand is the demand facing firm 1 after the quantity supplied by firm 2 is subtracted from the aggregate industry demand In the present case we subtract from the aggregate demand curve to obtain the residual demand curve facing firm 1 given by q1 10 p 3 7 p or its inverse p 7 q1 The most important observation to be made about Figure 63 is that the marginalrevenue curve derived from this residualdemand function M R1q1 7 2q1 is strictly positive for all output levels satisfying implying that the residual demand is elastic at this interval Therefore increasing p1 will only reduce the revenue of firm 1 This establishes the following claim Lemma 61 If the output capacity levels chosen in period 1 satisfy then the Nash equilibrium exhibits both firms choosing the marketclearing price in the second period Lemma 61 shows that given firms choices of output levels in the secondperiod price game firms will strategically choose to play the market price that clears the market at the given aggregate output level Page 115 The firstperiod game In the first period firms observe that the secondperiod price would be the marketclearing price Lemma 61 Therefore for each firm the firstperiodcapacitychoice problem is precisely the Cournotquantitychoice problem as formulated in Definition 24 Hence in the first period firms would choose the Cournot quantity levels Intuitively in the first period both firms know that the secondperiod price choices by both firms would be the price that clears the market for the firstperiod production levels This knowledge makes the firms firstperiodoutputchoice problem identical to firms output choices in a Cournot market structure as defined in Definition 61 Finally note that this illustration does not provide a complete proof for this statement since in Lemma 61 we assumed that the firms did not choose very high capacity levels in the first period In that respect Lemma 61 is not proven for output levels exceeding q1 q2 6 We refrain from proving that in order to avoid using mixed strategies in this book Also from time to time this result causes some confusion among students and researchers leading them to state that there is no reason for using Bertrand price competition anymore since the twoperiod capacityprice game would yield the same outcome as the Cournot market structure Note that this statement is too strong since it holds only for the particular twoperiod game analyzed in the present section 65 SelfEnforcing Collusion In this section we extend the basic static Cournot game to an infinitely repeated game in which firms produce output and collect profits in each period Although the analysis in this section is self contained the reader is urged to obtain some background on repeated games by reading section 23 One very important result will emerge from analyzing an infinitely repeated Cournot game namely that the outcome in which all firms produce the collusive output levels see the cartel analysis in subsection 541 constitutes a subgame perfect equilibrium for the noncooperative repeated Cournot game More precisely in subsection 612 we proved that under the Cournot market structure with two or more firms aggregate industry output exceeds the monopoly output level which equals the cartels total output level Moreover we showed that as the number of firms increases the output level increases and converges to the competitive output level Altogether firms have a lot to gain by colluding rather than competing under any market structure In this section we show that if the Cournot game is repeated infinitely then Page 11 the collusive output level can emerge as a noncooperative equilibrium The importance of this result is that it implies that observing an industry where production levels are limited and firms make strictly positive profits does not imply that the firms are engaged in any cooperative activities In fact what we show in this section is that the cooperative collusive output levels can be sustained as a noncooperative equilibrium In the subsection 651 we develop a simple Cournot duopoly model and analyze the incentives to collude among firms and the incentive for each firm to unilaterally deviate from collusion when the game is played only once Subsection 652 analyzes equilibrium outcome when the oneshot game is repeated infinitely 651 The oneshot game Consider the following basic oneshot Cournot game There are two firms denoted by i 1 2 We denote by qi the output level of firm i The demand facing the industry is p 1 q1 q2 Let denote the aggregate industryoutput level and assume that production is costless In the following subsubsections we quickly derive the already familiar Cournot duopoly equilibrium the collusion cooperative monopoly equilibrium and then the incentives to deviate from the cooperative outcome Duopoly Noncooperative behavior In view of Definition 61 in a Cournot market structure firm 1 maximizes π1 1 q1 q2q1 yielding a bestresponse function q1 q2 1 q22 and the equilibrium output levels where M stands for medium production level Hence Q 23 and p 13 implying that πi 19 The profits of the firms under duopoly are displayed in the second column and second row of Table 61 Firm 2 Firm 1 Table 61 Cooperation L Noncooperative Cournot duopoly M Defection from cooperation H Page 117 Collusion Cooperative behavior We assume that when the two firms collude they act as a cartel analyzed in subsection 541 Since the firms have identical technologies that exhibit constant returns to scale the present case is easy to analyze because under CRS there is no difference whether under collusion they operate one or two plants In any case the cartels profitmaximizing output is found by equating M RQ 1 2Q 0 MCi implying that Q 12 p 12 Hence equal division of output between the two colluding firms imply that qi L 14 where L stands for low output levels Thus as expected collusion implies that both firms restrict their output levels below the Cournot output levels The two firms equally divide the profit so πi pQ2 18 which is displayed in the first column and row in Table 61 Deviation from collusion Suppose that firm 2 plays the naive collusive output level q2 L We now show that in this oneshot game firm 1 can increase its profit by unilaterally increasing its output level To see that for given q1 14 firm 1 chooses q1 to max π 1 q1 14q1 yielding 0 34 2q1 Hence Thus if firm 2 does not deviate from q2 L firm 1 has the incentive to increase its output to a high level In this case Q 38 14 58 p 38 π1 964 and π2 332 both are displayed in the first column third row in Table 61 Equilibrium in the oneshot game The first part of the next proposition follows directly from equation 65 and also from Table 61 The second part follows from Definition 26 and Table 61 Proposition 63 In the oneshot game 1 there exists a unique CournotNash equilibrium given by q1 q2 M 13 2 the equilibrium outcome is Pareto dominated by the cooperative outcome q1 q2 L 14 Note that we use the Pareto criterion to refer only to the profit of firms thereby disregarding consumers welfare 652 The infinitely repeated game Suppose now that the two firms live forever The game proceeds as follows In each period t both firms observe what both firms played in Page 118 all earlier periods observe period t history as defined in Definition 211 and then play the oneshot game described in Table 61 That is in each period t each firm i chooses qit where i 12 and t 01 2 A strategy of firm i is a list of output levels chosen each period by firm i after the firm observed all the output levels chosen by each firm in all earlier periods see Definition 211 for a precise definition of a strategy in repeated games Let 0 ρ 1 be the discount factor Note that in perfect capital markets the discount factor is inversely related to the interest rate Let r denote the interest rate Then As r rises ρ falls meaning that future profits are less valuable today Following Assumption 21 we assume that the objective of each firm is to maximize the sum of present and discounted future profits given by where the values of πit are given in Table 61 The trigger strategy We restrict the discussion here to one type of strategies called trigger strategies meaning that in every period τ each player cooperates playing qiτ L as long as all players including himself cooperated in all periods t 1 τ 1 see Definition 211 for a precise definition However if any player deviated in some period then player i plays the noncooperative duopoly strategy forever That is qit M for every t ττ 1τ 2 Formally let us restate Definition 212 for the present game Definition 64 Player i is said to be playing a trigger strategy if for every period τ τ 1 2 In other words firm i cooperates by restricting its output as long as all firms restrict their output levels in earlier periods However if any firm deviates even once then firm i produces the static CournotNash duopoly output level forever Equilibrium in trigger strategies We now seek to investigate under what conditions playing trigger strategies constitutes a subgame perfect equilibrium see Definition 210 It Page 119 turns out that for a small discount factor a firm may benefit by deviating from the cooperative output level thereby collecting a temporary high profit by sacrificing the extra future profits generated by cooperation However for a sufficiently large discount factor we can state the following proposition Proposition 64 If the discount factor is sufficiently large then the outcome where both firms play their trigger strategies is a SPE Formally trigger strategies defined in Definition 64 constitute a SPE if ρ 917 Proof We look at a representative period call it period τ and suppose that neither firm has deviated in periods t 1 τ 1 Then if firm 1 deviates and plays q1τ H the best response to q2τ L Table 61 shows that π1τ 964 18 However given that firm 1 deviates firm 2s equilibrium strategy calls for playing q2t M for every Hence the period τ 1 sum of discounted profits of firm 1 for all periods is Note that we used the familiar formula for calculating the present value of an infinite stream of profits given by Hence if firm 1 deviates in period τ its sum of discounted profits is However if firm 1 does not deviate in period τ then both firms continue producing the collusive output yielding Comparing 623 with 624 yields the conclusion that deviation is not profitable for firm 1 if ρ 917 As we noted in the proof of Proposition 25 to prove subgame perfection we need to show that each firm would find it profitable to respond with deviation when it realizes that deviation occurred in an earlier period as stated in the definition of the trigger strategy described in Definition 64 That is we still need to show that a firm would produce a level of M forever once either firm deviated in an earlier period In the language of game theorists we need to show that the trigger strategy is the best response even if the game drifts off the equilibrium path However Definition 64 implies that if firm j deviates then firm j would produce M in all future periods Then Table 61 shows that firm is best response to firm js playing M is to play M Hence the trigger strategies defined in Definition 64 constitute a SPE Page 120 Discussion of trigger strategies and extensions The purpose of section 65 was to demonstrate that in an infinitely repeated game the set of oligopoly equilibria is larger than that of a oneshot game and includes cooperative outcomes in addition to the familiar noncooperative outcome Readers who wish to learn more about cooperation in oligopolistic market structures are referred to Abreu 1986 Friedman 1971 1977 Green and Porter 1984 Segerstrom 1988 Tirole 1988 chap 5 and more recent books on game theory noted in the references to chapter 2 We conclude our analysis of dynamic collusion with two remarks a We have not discussed what would happen to our cooperative equilibrium when we increase the number of firms in the industry Lambson 1984 has shown that under general demand conditions the cooperation continues to hold as long as the demand for the product increases at the same rate as the number of firms The intuition behind this result is as follows If the number of firms grows over time but the demand stays constant then the future profit of each firm would drop implying that firms would have a stronger incentive to deviate from the collusive output level Hence in such a case collusion is less likely to be sustained b Another natural question to be asked is how booms and recessions affect the possibility of collusion among firms Rotemberg and Saloner 1986 analyze collusion under stochastic demand The problem they investigate is whether collusion is more sustainable during booms a high realization of the demand than during recessions a low demand realization 66 International Trade in Homogeneous Products In this section we analyze two issues related to international trade in homogeneous products Subsection 661 demonstrates the possibility that countries sell homogeneous products below cost in other countries Subsection 662 evaluates how the formation of customs unions and free trade agreements affect international trade in homogeneous products 661 Reciprocal dumping in international trade An application of the Cournot equilibrium for international trade is given in Brander and Krugman 1983 Suppose that there are two identical trading countries indexed by k k 1 2 The demand schedule in each country is given by pkQk a bQk where Q is the sum of local production and import In each country there is one firm producing a homogeneous product that is sold both at home and abroad To keep this example simple assume that production is costless that is c 0 Page 121 The two countries are separated by an ocean and therefore shipping the good across the continents is costly Also assume that the transportation cost is paid by the exporting firm Let τ denote the perunit international transportation cost and let qk denote the production level of the firm located in country k k 1 2 Since each firm sells both at home and abroad the output of firm k is decomposed into home local sales denoted by and foreign export sales denoted by Therefore the total output sold in country 1 is and the total output sold in country 2 is The profit of each firm is the revenue collected in each country minus the cost of production assumed to be zero minus export transportation cost Formally the profit of the firm located in country 1 is The profit of the firm located in country 2 is The firstorder conditions for 625 are Notice that the two firstorder conditions are independent in the sense that foreign sales does not appear in the first condition and home sales does not appear in the second This follows from our particular use of the linear cost structure In general when the cost function is nonlinear the two conditions would not be independent The firstorder conditions for 626 Using this special case we can solve for the Cournot equilibrium output levels for each country separately In this case 65 implies that for firm k k 1 2 Note that as transportation becomes more costly τ increases the share of domestic sales increases in each country whereas the level of export declines Also as τ increases pk increases Page 122 Dumping One of the major rules of GATT General Agreement on Tariffs and Trade is that dumping is prohibited Before we define dumping we need to distinguish between two types of prices used in international transactions a FOB price freeonboard meaning the price received by the producer when the product leaves the plant This price does not include the payments for transportation and insurance b CIF price costinsurancefreight which includes all transportation as well as insurance costs If we assume away dealers which would make the CIF price even higher the consumer pays the CIF price whereas the exporter receives the FOB price per unit of export Brander and Krugman 1983 use the term dumping to describe a situation where the FOB export price is lower than the price charged for domestic sales Formally in the present model Thus each firm in each country dumps the product in the other country by subsidizing the transportation cost Another commonly used definition of dumping is when a firm sells abroad at a price below cost This does not happen in the present model Finally note that for this problem the Cournot market structure generates inefficient trade since the world could save the transportation cost if each firm sells only in its home country However in general making each firm a monopoly in its own country would generate the other familiar inefficiencies 662 Homogeneous products and preferential trade agreements among countries There are three general types of trade agreements among countries 1 the freetrade agreement FTA which is an agreement among countries to eliminate trade barriers among the member countries but under which each country is free to set its own trade restrictions against trade with nonmember countries 2 the customs union CU which is an agreement among countries to eliminate tariffs on goods imported from other member countries of the union and to set a uniform trade policy regarding nonmember countries and 3 the common market CM where in addition to the elimination of tariffs among member countries and in addition to the common tariff policy toward nonmembers there is a free movement of factors of production among member countries Formal analyses of these agreements were first given by Viner Meade and Vanek and the interested reader is referred to surveys of literature Page 123 given in Corden 1984 and Vousden 1990 or in almost any elementary book on international trade Consider the following world There are three countries the European Community EC the Far East FE and Israel IL Assume that IL is a small country thus it cannot affect the world prices Only FE and EC produce carpets that are imported by IL Assume that carpets cannot be produced in IL We further assume that ILs demand for imported carpets is given by pIL a Q where Q denotes the quantity demanded and pIL is the domestic tariffinclusive price Assume that initially period 0 IL sets a uniform tariff of t per carpet irrespective of where the carpets are imported from Then in period 1 assume that IL signs a freetrade agreement FTA with EC Period 0 IL levies a uniform tariff on carpets We denote by pEC the price of a carpet charged by ECs producers and by PFE the price charged by FEs producers Hence with a uniform tariff of t the price paid by ILs consumers for carpets imported from EC is and the price paid for carpets imported from FE is We make the following assumption Assumption 61 The export price of carpets in EC exceeds the export price in FE Formally pEC pFE Figure 64 illustrates ILs demand for imported carpets and the prices with and without the tariff on carpets imported from EC and FE Figure 64 shows that IL will import from the cheapest supplier which is Figure 64 ILs import level under a uniform tariff FE so that the import level would be Q0 In this case the governments revenue from importtariff collection would be G0 tQ0 The ILs Page 124 consumer surplus see subsection 323 for a definition is given by Also note that We define ILs social welfare as the sum of consumer surplus plus ILs government revenue from tariff collection Note that in modeling international trade it is very important not to forget the existence of governments revenue and to assume that the government returns the tariff revenue to consumers in a lumpsum fashion or by other services Hence implying that Note that the last step in 629 uses the mathematical identity that Equation 629 shows that the welfare of country IL decreases with the tariff rate t and with FEs price of carpets Period 1 IL signs a freetrade agreement with the EC Now suppose that IL signs a FTA with EC so that the tariff on carpets imported from EC is now set to zero whereas the tariff on imports from FE remains the same at the level of t per unit Figure 65 illustrates that IL switches from importing from FE to importing from only EC for a price of Given that the price of carpets drops in IL the Figure 65 ILs import under the FTA quantity of imported carpets increases to Notice that although ILs consumer price of carpets has decreased IL now buys carpets from the more expensive source Page 125 Under the FTA since all the imports are from EC the government collects zero revenue that is G1 0 Hence ILs social welfare equals ILs consumer surplus That is W1 CS1 The consumers surplus is illustrated in Figure 65 and is calculated to be Welfare analysis of the freetrade agreement We now analyze whether IL gains from the FTA with EC Comparing 629 and 630 we see that the FTA improves ILs welfare if W1 W0 That is or Therefore Proposition 65 A freetrade agreement between IL and EC is more likely to be welfare improving for IL when a the initial uniform tariff is high and b when the difference in prices between the two foreign exporters is small that is when pEC is close to PFE We conclude this analysis with a graphic illustration of the gains and loss from the FTA Figure 66 illustrates the welfare implication of ILs signing the FTA with EC In Figure 66 the area denoted by φ Figure 66 The welfare effects of the freetrade agreement measures ILs consumer surplus prior to signing the FTA The sum of the areas β δ measures ILs government tariff revenue prior to signing the agreement Hence ILs welfare prior to signing the agreement is W0 φ β β Page 126 In Figure 66 the sum of the areas φ β γ measures ILs consumer surplus after the FTA is signed Since there are no tariff revenues after the FTA all carpets are imported from the EC the welfare of IL after the FTA is W1 φ β λ Altogether the welfare change resulting from signing the FTA is given by Defintion 65 The change in consumer surplus due to the increase in the consumption of the imported good area γ in Figure 66 is called the tradecreation effect of the FTA The change in the importing countrys expenditure due to the switch to importing from the more expensive country area δ in Figure 66 is called the tradediversion effect of the FTA Thus the importing country gains from the FTA if the positive tradecreation effect associated with the increase in the import level dominates the negative tradediversion effect associated to switching to importing from the more expensive source 67 Appendix Cournot Market Structure with Heterogeneous Firms In this appendix we extend the analysis conducted in Subsection 612 and solve for the Cournot marketstructure equilibrium when there is a large number of firms with different cost functions Following Bergstrom and Varian 1985 we introduce a method for calculating a CournotNash equilibrium output level without resorting to solving N firstorder conditions for the equilibrium N output levels In a Cournot market structure with N firms each with a unit cost of each firm i chooses its output qi that solves yielding assuming for all i a firstorder condition Now instead of solving N equations N firstorder conditions for N output levels we solve for the aggregate production level by rewriting the firstorder conditions in the form of Page 127 Summing over all qi i 1 N yields Hence the Cournot equilibrium aggregate industry output and market price are given by Hence Proposition 66 In an industry where firms have constant unit costs if in a Cournot equilibrium all firms produce strictly positive output levels then the Cournot aggregate industry equilibrium output and price levels depend only on the sum of the firms unit costs and not on the distribution of unit costs among the firms The result stated in Proposition 66 is important since it implies that under constant unit costs industry output price and hence total welfare can be calculated by using the sum of firms unit costs without investigating the precise cost distribution among firms Moreover the proof of Proposition 66 does not rely on linear demand and therefore also applies to nonlinear demand functions We conclude this appendix by illustrating a simple application of Proposition 66 Consider an industry consisting of two type of firms highcost and lowcost firms Suppose that there are highcost firms with a unit production cost given by cH and lowcost firms with a unit production cost given by cL where Substituting into 632 yields Hence the Cournot output and price equilibrium levels depend only on HcH LcL The advantage of learning this method for calculating Cournot equilibrium outcomes becomes clear in the case where there is an entry or exit of some firms For example suppose we observe that three additional low cost firms have joined the industry Then the new Cournot equilibrium industry output and price can be immediately calculated by replacing HcH LcL with HcH L 3cL in 633 Page 128 68 Exercises 1 Two firms produce a homogeneous product Let p denote the products price The output level of firm 1 is denoted by q1 and the output level of firm 2 by q2 The aggregate industry output is denoted by Q The aggregate industry demand curve for this product is given by p a Q Assume that the unit cost of firm 1 is c1 and the unit cost of firm 2 is c2 where α c2 c1 0 Perform the following a Solve for a competitive equilibrium see Definition 42 on page 65 Make sure that you solve for the output level of each firm and the market price b Solve for a Cournot equilibrium see Definition 61 on page 99 Make sure that you solve for the output level of each firm and the market price c Solve for a sequentialmoves equilibrium see Section 62 on page 104 assuming that firm 1 sets its output level before firm 2 does d Solve for a sequentialmoves equilibrium assuming that firm 2 sets its output level before firm I does Is there any difference in market shares and the price level between the present case and the case where firm 1 moves first Explain e Solve for a Bertrand equilibrium see Definition 62 on page 108 Make sure that you solve for the output level of each firm and the market price 2 In an industry there are N firms producing a homogeneous product Let qi denote the output level of firm i i 1 2 N and let Q denote the aggregate industry production level That is Assume that the demand curve facing the industry is p 100 Q Suppose that the cost function of each firm i is given by Solve the following problems a Suppose that the number of firms in the industry N is sufficiently small so that all the N firms make abovenormal profits Calculate the output and profit levels of each firm in a Cournot equilibrium b Now assume that firms are allowed to enter or the exit from the industry Find the equilibrium number of firms in the industry as a function of F Hint Equate a firms profit level that you found earlier to zero and solve for N 3 Consider a threeperiod version of the sequentialmoves equilibrium analyzed in section 62 Assume that the market inverse demand curve is Page 129 given by p 120 Q and suppose that there are three firms that set their output levels sequentially firm 1 sets q1 in period 1 firm 2 sets q2 in period 2 and firm 3 sets q3 in period 3 Then firms sell their output and collect their profits Solve for the sequentialmoves equilibrium assuming that production is costless Make sure that you solve for the output level of each firm and the market price 4 Two firms compete in prices in a market for a homogeneous product In this market there are N 0 consumers each buys one unit if the price of the product does not exceed 10 and nothing otherwise Consumers buy from the firm selling at a lower price In case both firms charge the same price assume that N2 consumers buy from each firm Assume zero production cost for both firms a Find the Bertrand equilibrium prices for a singleshot game assuming that the firms choose their prices simultaneously b Now suppose that the game is repeated infinitely Let ρ denote the timediscount parameter Propose trigger Price strategies for both firms yielding the collusive prices of 10 10 each period Calculate the minimal value of ρ that would enforce the trigger price strategies you proposed c Now suppose that the unit production cost of firm 2 is 4 but the unit cost of firm 1 remained zero Find the Bertrand equilibrium prices for the singleshot game d Assuming the new cost structure propose trigger price strategies for both firms yielding the collusive prices of 10 10 each period and calculate the minimal value of ρ that would enforce the trigger price strategies you propose e Conclude whether it is easier for firms to enforce the collusive prices when there is symmetric industry cost structure or when the firms have different cost structures Explain 5 Consider the freetrade agreement model analyzed in subsection 662 Suppose that the world consists of three countries denoted by A B and C Country A imports shoes from countries B and C and does not have local production of shoes Let the export shoe prices of countries B and C be given by pB 60 and pC 40 Also suppose that initially country A levies a uniform import tariff of t 10 per each pair of imported shoes Answer the following questions a Suppose that country A signs a FTA with country B Does country A gain or lose from this agreement Explain b Suppose now that initially the export price of shoes in country C is pC 5001 Under this condition will country A gain or lose from the FTA Explain 6 In a market for luxury cars there are two firms competing in prices Each firm can choose to set a high price given by pH or a low price Page 130 given by pL where The profit levels of the two firms as a function of the prices chosen by both firms is given in Table 62 The rules of this twostage market game are as follows In the first Firm 2 pH pL Firm 1 pH 100 100 0 120 pL 120 0 70 70 Table 62 Meet the competition clause stage firm 1 sets its price In the second stage firm 1 cannot reverse its decision whereas firm 2 observes p1 and then chooses Then the game ends and each firm collects its profit according to Table 62 a Formulate the game in extensive form Definition 27 on page 24 by drawing the game tree and solve for the subgame perfect equilibrium Definition 210 on page 27 for this game b Suppose now that firm 1 offers its consumers to match its price with the lowest price in the market the socalled meet the competition clause Solve for the subgame perfect equilibrium for the modified game Hint Modify the game to three stages allowing firm 1 to make a move in the third stage only in the case where it chose pH in the first stage and firm 2 chose pL in the second stage 7 This problem is directed to highly advanced students only Suppose there are N 2 firms that set their output sequentially as described in section 62 Suppose that all firms have identical unit costs given by c and suppose that the market inverse demand curve facing this industry is given by p a Q where and a Solve for the sequentialmoves equilibrium by showing that the output level of the firm that moves in period i i 1 N is given by b Show that the aggregate equilibriumoutput level is given by c Conclude what happens to the aggregate industryoutput level when the number of firms and periods increases with no bounds ie when Page 131 69 References Abreu D 1986 Extremal Equilibria of Oligopolistic Supergames Journal of Economic Theory 39 191225 Bergstrom T and H Varian 1985 When Are Nash Equilibria Independent of the Distribution of Agents Characteristics Review of Economics Studies 52 715718 Bertrand J 1883 Reviews of Théories Mathematique de la Richesse Sociale by Léon Walras and of Recherches sur les Principles Mathematiques de la Théorie des Richesses by Augustin Cournot Journal des Savants 67 499508 Bork R 1978 The Antitrust Paradox New York Basic Books Brander J and P Krugman 1983 A Reciprocal Dumping Model of International Trade Journal of International Economics 15 313321 Corden M 1984 The Normative Theory of International Trade In Handbook of International Economics edited by R Jones and P Kenen Amsterdam NorthHolland Cournot A 1929 1838 Researches into the Mathematical Principles of the Theory of Wealth Translated by Nathaniel Bacon New York Macmillan Edgeworth F 1925 1897 The Pure Theory of Monopoly In Papers Relating to Political Economy edited by F Edgeworth London Macmillan Friedman J 1971 A Noncooperative Equilibrium for Supergames Review of Economic Studies 38 112 Friedman J 1977 Oligopoly and the Theory of Games Amsterdam NorthHolland Green E and R Porter 1984 Noncooperative Collusion Under Imperfect Price Information Econometrica 5287100 Konow J 1994 The Political Economy of Heinrich yon Stackelberg Economic Inquiry 32 146 165 Kreps D and J Scheinkman 1983 Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes Bell Journal of Economics 14 326337 Lambson V 1984 SelfEnforcing Collusion in Large Dynamic Markets Journal of Economic Theory 34 282291 Rotemberg J and G Saloner 1986 SupergameTheoretic Model of Business Cycles and Price Wars During Booms American Economic Review 76 390407 Segerstrom P 1988 Demons and Repentance Journal of Economic Theory 45 3252 Shubik M 1987 Cournot Antoine Augustin In The New Palgrave Dictionary of Economics edited by J Eatwell M Milgate and P Newman New York The Stockton Press Page 132 von Stackelberg H 1934 Marktform und Gleichgewicht Market structure and equilibrium Vienna SpringerVerlag Tirole J 1988 The Theory of Industrial Organization Cambridge Mass MIT Press Vousden N 1990 The Economics of Trade Protection Cambridge Cambridge University Press Page 133 Chapter 7 Markets for Differentiated Products You can have it any color you want as long as its black Attributed to Henry Ford In this chapter we analyze oligopolies producing differentiated products Where in chapter 6 consumers could not recognize or did not bother to learn the producers names or logos of homogeneous products here consumers are able to distinguish among the different producers and to treat the products brands as close but imperfect substitutes Several important observations make the analysis of differentiated products highly important 1 Most industries produce a large number of similar but not identical products 2 Only a small subset of all possible varieties of differentiated products are actually produced For example most products are not available in all colors 3 Most industries producing differentiated products are concentrated in the sense that it is typical to have two to five firms in an industry 4 Consumers purchase a small subset of the available product varieties This chapter introduces the reader to several approaches to modeling industries producing differentiated products to explain one or more of these observations Page 134 Product differentiation models are divided into two groups nonaddress models and address location models Figure 71 illustrates the logical connections among the various approaches The nonaddress Figure 71 Approaches to modeling differentiatedproducts industries approach displayed on the left main branch of Figure 71 is divided into two categories a fixed number of differentiated brands models and endogenously determined variety models The fixed number of brands approach is analyzed in section 71 Simple Models for Differentiated Page 135 Products where we analyze and compare quantity and price competition between the two differentiatedbrands producers Basic definitions for the degrees of product differentiation are provided and utilized in the two types of market structures Section 72 Monopolistic Competition analyzes a general equilibrium environment where free entry is allowed so the number of brands in an industry is determined in the model itself We assume that the economy is represented by a single consumer whose preferences exhibit love for variety of differentiated brands and that firms technologies exhibit returns to scale together with fixed cost of production Assuming free entry of firms enables us to compute the equilibrium variety of differentiated brands The monopolistic competition approach proves to be extremely useful in analyzing international markets which is discussed in subsection 722 The address location approach displayed on the right main branch of Figure 71 is analyzed in section 73 Location Models This approach provides an alternative method for modeling product differentiation by introducing location or addresses into consumers preferences that measure how close the brands actually produced are to the consumers ideal brands This approach is useful to model heterogeneous consumers who have different tastes for the different brands Together sections 72 and 73 discuss the two major approaches to product differentiation the non address approach and the address approach respectively see a discussion in Eaton and Lipsey 1989 The major difference between the approaches is that in the nonaddress approach all consumers gain utility from consuming a variety of products and therefore buy a variety of brands such as a variety of music records of movies of software of food etc In contrast the address location approach each consumer buys only one brand such as one computer one car or one house but consumers have different preferences for their most preferred brand A third approach to product differentiation not discussed in this chapter is found in Lancaster 1971 Lancasters characteristics approach assumes that each product consists of many characteristics such as color durability safety strength in choosing a specific brand the consumer looks for the brand that would yield the most suitable combinations of the products characteristics Finally a reader interested in applications of product differentiation to the readytoeat cereals industry is referred to Scherer 1979 and Schmalensee 1978 71 Simple Models for Two Differentiated Products Consider a twofirm industry producing two differentiated products indexed by i 1 2 To simplify the exposition we assume that production is costless Following Dixit 1979 and Singh and Vives 1984 we as Page 136 sume the following inverse demand structure for the two products Thus we assume that that there is a fixed number of two brands and that each is produced by a different firm facing an inverse demand curve given in 71 The assumption of β2 γ2 is very important since it implies that the effect of increasing q1 on p1 is larger than the effect of the same increase in q2 That is the price of a brand is more sensitive to a change in the quantity of this brand than to a change in the quantity of the competing brand A common terminology used to describe this assumption is to say that the ownprice effect dominates the crossprice effect The demand structure exhibited in 71 is formulated as a system of inverse demand functions where prices are functions of quantity purchased In order to find the direct demand functions quantity demanded as functions of brands prices we need to invert the system given in 71 The appendix section 74 shows that How to measure the degree of brand differentiation We would now like to define a measure for the degree of product differentiation Definition 71 The brands measure of differentiation denoted by δ is 1 The brands are said to be highly differentiated if consumers find the products to be very different so a change in the price of brand j will have a small or negligible effect on the demand for brand i Formally brands are highly differentiated if δ is close to 0 That is when hence 2 The brands are said to be almost homogeneous if the crossprice effect is close or equal to the ownprice effect In this case prices of all brands will have strong effects on the demand for each brand more precisely if an increase in the price brand j will increase the Page 137 demand for brand i by the same magnitude as a decrease in the price of brand i that is when δ is close to 1 or equivalently when hence Figure 72 illustrates the relationships between the the demand parameters β and γ as described in Definition 71 In Figure 72 a hori Figure 72 Measuring the degree of product differentiation zontal movement toward the diagonals implies that the products are becoming more homogeneous In contrast a movement toward the center is associated with the products becoming more differentiated 711 Quantity game with differentiated products We now solve for the prices and quantity produced under the Cournot market structure where firms choose quantity produced as actions Just as we did in solving a Cournot equilibrium for the homogeneous products case we look for a Nash equilibrium in firms output levels as defined in Definition 61 on page 99 Assuming zero production cost using the inverse demand functions given in 71 we note that each firm i takes qj as given and chooses qi to The firstorder conditions are given by yielding Page 138 best response functions given by Figure 73 illustrates the bestresponse functions in the q1 q2 space Notice that these functions are similar to the ones obtained for the Cournot game with homogeneous products illustrated in Figure 61 Notice that as the products are more homogeneous the best Figure 73 Bestresponse functions for quantity competition in differentiated products response function becomes steeper thereby making the profitmaximizing output level of firm i more sensitive to changes in the output level of firm j due to stiffer competition In contrast as the bestresponse function becomes constant zero sloped since the products become completely differentiated Solving the bestresponse functions 74 using symmetry we have that Clearly as γ increases the products are leas differentiated the individual and aggregate quantity produced the prices and the profits all decline Hence Proposition 71 In a Cournot game with differentiated products the profits of firms increase when the products become more differentiated Page 139 The importance of Proposition 71 is that it can explain why firms tend to spend large sums of money to advertise their brands because firms would like the consumers to believe that the brands are highly differentiated from the competing brands for the purpose of increasing their profits In other words differentiation increases the monopoly power of brandproducing firms 712 Price game with differentiated products We now solve for the prices and quantity produced under the Bertrand market structure where firms choose prices as their actions Just as we did in solving for a Bertrand equilibrium for the homogeneous products case we look for a Nash equilibrium in firms prices as defined in Definition 62 on page 108 for the homogeneous product case Using the direct demand functions given in 72 each firm i takes pj as given and chooses pi to The firstorder conditions are given by yielding bestsresponse functions given by The bestresponse functions are drawn in Figure 74 You have probably Figure 74 Bestresponse functions for price competition in differentiated products Page 140 noticed that there is something different in Figure 74 compared with what is in Figure 73 In price games the bestresponse functions are upward sloping meaning that if one firm raises its price the other would respond by raising its price as well Well this discovery deserves a definition Bulow Geanakoplos and Klemperer 1985 Definition 72 1 Players strategies are said to be strategic substitutes if the bestresponse functions are downward sloping 2 Players strategies are said to be strategic complements if the bestresponse functions are upward sloping Note that this terminology may be misleading since there is no relationship between this definition and whether goods are substitutes or complements in consumption Definition 72 implies that in a quantity game the quantities are strategic substitutes whereas in a price game prices are strategic complements Solving 77 yields for i 1 2 The profit levels decline when the products become less differentiated γ increases In the limit when γ β the products become homogeneous and the profits drop to zero as in the Bertrand equilibrium for homogeneous products analyzed in section 62 Hence Proposition 72 In a Bertrand game with differentiated products the profits of firms increase when the products become more differentiated As with the Cournot case product differentiation increases the monopoly power of brandproducing firms by loosening up price competition among the brandproducing firms 713 Cournot versus Bertrand in differentiated products Which market structure a Cournot or a Bertrand would yield a higher market price How would changing the degree of product differentiation affect the relative difference between the two market structure outcomes As you may expect the price under Bertrand is indeed lower than it is under the Cournot market structure Formally comparing 75 with 78 yields Page 141 Thus Proposition 73 In a differentiated products industry 1 The market price under Cournot is higher than it is under Bertrand Formally 2 The more differentiated the products are the smaller the difference between the Cournot and Bertrand prices Formally 3 This difference in prices is zero when the products become independent Formally The intuition behind Proposition 73 given in Vives 1985 is as follows Under Cournot market structure each firm expects the other firm to hold its output level constant Hence each firm would maintain a low output level since it is aware that a unilateral output expansion would result in a drop in the market price In contrast under the Bertrand market structure each firm assumes that the rival firm holds its price constant hence output expansion will not result in a price reduction Therefore more output is produced under the Bertrand market structure than under the Cournot market structure Cheng 1985 provides some additional graphical intuition for the differences between the market outcomes obtained under the two market structures 714 Sequentialmoves price game Consider a twoperiod pricesetting sequential game that is similar to the sequentialmoves quantity game described in section 62 but here we let firms set prices rather than quantity produced In order to have some fun let us take a specific numerical example for the demand system given in 72 For this particular example 78 implies that the singleperiod game Bertrand prices and profit levels are and Following the same logical steps as those in section 62 we look for a SPE in prices where firm 1 sets its price before firm 2 Thus in the first period firm 1 takes firm 2s bestresponse function 77 as given and chooses p1 that solves Page 142 The firstorder condition is Therefore hence Substituting into 710 yields that and q2 114 Hence and Why do we bother to go over this exercise under a price game Well the following proposition yields a rather surprising result concerning the relationship between firms profit levels and the order of moves Proposition 74 Under a sequentialmoves price game or more generally under any game where actions are strategically complements 1 Both firms collect a higher profit under a sequentialmoves game than under the singleperiod Bertrand game Formally for i 1 2 2 The firm that sets its price first the leader makes a lower profit than the firm that sets its price second the follower 3 Compared to the Bertrand profit levels the increase in profit to the first mover the leader is smaller than the increase in profit to the second mover the follower Formally It this amazing What we have learned from this example is that being the first to move is not always an advantage Here each firm would want the other firm to make the first move The intuition behind this result is as follows When firm 1 sets its price in period 1 it calculates that firm 2 will slightly undercut p1 in order to obtain a larger market share than firm 1 This calculation puts pressure on firm 1 to maintain a high price to avoid having firm 2 set a very low market price Hence both firms set prices above the static Bertrand price levels Now firm 1 always makes a lower profit than firm 2 since firm 2 slightly undercuts firm 1 and captures a larger market share Finally note that we could have predicted that the profit of firm 1 will increase beyond the static Bertrand profit level even without resorting to the precise calculations Using a revealed profitability argument we can see clearly that firm 1 can always set and make the same profit as under the static Bertrand game However given that firm 1 chooses a different price its profit can only increase Finally part 1 of Proposition 74 reveals the major difference between the price sequentialmoves game and the quantity sequentialmoves game analyzed in section 62 Here the profit of firm 2 the followers is higher under the sequentialmoves price game than its profit under the static Bertrand game In contrast under the sequentialmoves quantity game the followers profit is lower than it is under the static Cournot game Page 143 72 Monopolistic Competition in Differentiated Products In this section we analyze a monopolisticcompetition environment Chamberlin 1933 Our major goal is to calculate the equilibrium number of differentiated brands produced by the industry The main features of this environment are that 1 consumers are homogeneous have identical preferences or can be represented by a single consumer who loves to consume a variety of brands Thus this model better describes markets in which consumers like to consume a large variety of brandssuch as a variety of music records of video of clothes and of moviesrather than markets for cars where most individuals consume at most one unit 2 there is an unlimited number of potentially produced brands and 3 free entry of new brandproducing firms It should be pointed out that this model is a general equilibrium one Unlike the partial equilibrium models the general equilibrium model is one where consumers demand is derived from a utility maximization where the consumers income is generated from selling labor to firms and from owning the firms Subsection 721 analyzes a singleeconomy monopolistic competition and subsection 722 extends the model to two open economies 721 The basic model We analyze here a simplified version of Dixit and Stiglitz 1977 Consider an industry producing differentiated brands indexed by i 1 2 3 N where N is an endogenously determined number of produced brands We denote by the quantity producedconsumed of brand i and by pi the price of one unit of brand i Consumers In this economy there is a single representative consumer whose preferences exhibit the lovefor variety property Formally the utility function of the representative consumer is given by a constantelasticityofsubstitution CES utility function This type of utility function exhibits love for variety since the marginal utility of each brand at a zero consumption level is infinite That is Page 144 In addition Figure 75 illustrates that the indifference curves are convex to the origin indicating that the consumers like to mix the brands in their consumption bundle Also note that the indifference curves touch Figure 75 CES indifference curves for N 2 the axes therefore making it possible for the consumers to gain utility even when some brands are not produced hence not consumed We use the word representative consumer for this utility function since in reality individual consumers do not purchase the entire variety of products Sattinger 1984 proposed a method for aggregating individuals who purchase a single brand into aggregate market demand facing all the brandproducing firms Finally the consumers income denoted by I is composed of the total wages paid by the producing firms plus the sum of their profits if any We denote by πiqi the profit of the firm producing brand i We also normalize the wage rate to equal 1 so all monetary values pi I and πi are all denominated in units of labor Hence the consumers maximize their utility 712 subject to a budget constraint given by We form the Lagrangian Page 145 The firstorder condition for every brand i is Thus the demand and the price elasticity ηi for each brand i are given by Finally note that we assumed that A is a constant However λ is not reply a constant but a function of all prices and N This procedure would be right had we assumed a continuum of brads indexed on the interval In this case a rise in the price of a single brand would not have an effect on consumers expenditure and hence on θ The continuum version of 712 should be written as However in an attempt to avoid using integrals in this book we provide the present approach as a good approximation for the continuous case Brandproducing firms Each brand is produced by a single firm All potential firms have identical technologies identical cost structure with increasing returns to scale IRS technologies Formally the total cost of a firm producing qi units of brand i is given by Defining a monopolisticcompetition market structure Definition 73 The triplet is called a Chamberlinian monopolisticcompetition equilibrium if 1 Firms Each firm behaves as a monopoly over its brand that is given the demand for brand i 714 each firm i chooses to 2 Consumers Each consumer takes his income and prices as given and maximizes 712 subject to 713 yielding a system of demand functions 714 Page 146 ATCiqi MCiqi Figure 76 Decreasing averagecost technology 3 Free entry Free entry of firms brands will result in each firm making zero profits for all i 12 N 4 Resource constraint Labor demanded for production equals the total labor supply Definition 73 can be easily interpreted using Figure 76 The demand facing each existing brand producing firm depends on the total number of brands in the industry N When N increases the demand facing each brandproducing firm shifts downward reflecting the fact that consumers partially substitute higher consumption levels of each brand with a lower consumption spread over a large number of brands Therefore free entry increases the number of brands until the demand facing each firm becomes tangent to the firms average cost function At this point each existing brandproducing firm makes zero profit and entry stops The equilibrium condition in which demand becomes tangent to the average cost of each firm is known as Chamberlins tangency condition Two important observations follow from the tangency condition displayed in Figure 76 First in equilibrium the price of each brand equals average cost Second in equilibrium all brandproducing firms produce on the downward sloping part of the average cost curve Thus firms do not minimize average cost under a monopolisticcompetition market structure Page 147 Solving for a monopolisticcompetition equilibrium A firms profitmaximization problem item I of Definition 73 is the already familiar monopolys problem analyzed in chapter 5 In that chapter we showed that if a monopoly produces a strictly positive amount of output then the monopolys price would satisfy Hence the equilibrium price of each brand is given by twice the marginal cost The zeroprofit condition item 3 of Definition 73 implies that Hence We are left to find how many brands will be produced in this economy The resourceconstraint condition item 4 of Definition 73 implies that NF cFc L Hence N L2F Altogether we have it that Proposition 75 1 In a monopolistic competition equilibrium with strictly positive fixed and marginal cost only a finite number of brands will be produced The equilibrium is given by 2 When the fixed cost is large there will be a low variety of brands but each brand will be producedconsumed in a large quantity When the fixed cost is low there will be a large variety of brands and each will be producedconsumed in a small quantity 722 Monopolistic competition in international markets In the late 1970s trade theorists began applying the theory of monopolistic competition to international trade see Helpman and Krugman 1985 The major motivation was that the neoclassical international trade theory failed to explain the data showing that most international trade consists of trade with similar products intraindustry trade rather of very different products interindustry trade as predicted by the traditional factorproportion theory That is the application of monopolistic competition was needed in order to explain why countries trade in similar products There are two mutually dependent ways for explaining gains from trade under increasingreturns production technologies a trade Page 148 increases specialization thereby enabling firms to produce at a higher scale and therefore at a lower average cost and b trade increases the world variety of brands facing each consumer in each country Consider a twocountry world economy in which each country is identical to the one analyzed above Under autarky no trade each country is described by Proposition 75 Our first question is what would happen to the patterns of production and consumption when the two countries start trading move to a freetrade regime When the world is integrated into a single large economy the labor resource and the number of consumers basically doubles In view of the equilibrium described in Proposition 75 there will be no change in brand prices and the level of production of each brand However the number of brands under free trade will double and become Nf LF 2Na where f and a denote equilibrium values under free trade and under autarky respectively Also note that since the quantity produced of each brand remains unchanged but the entire population has doubled under free trade each consumer country consumes onehalf of the world production F2c Our second question is whether there are gains from trade given that we found that the consumption level of each brand has decreased to onehalf the autarky level while the number of brands has doubled In order to answer that we should calculate the equilibrium utility levels under autarky and under free trade Thus Hence each consumer in each country gains from trade The intuition is quite simple Comparing point a with point f in Figure 75 shows that a consumer is always better off if the variety doubles despite the decline in the consumption level of each brand We conclude our analysis of the gains from trade with two remarks First we have shown that under monopolistic competition free trade yields a higher welfare level than autarky However Gros 1987 has shown that countries may benefit from imposing some import tariff on foreign produced brands Second let us note that we have shown there are gains from trade when there is only one industry producing differentiated brands Chou and Shy 1991 have shown that the gains from trade in monopolistic competition extend to the case where some industries produce nontraded brands however the remote possibility that trade may reduce the welfare of all countries Pareto inferior trade remains Page 149 73 Location Models In this section we present models in which consumers are heterogeneous That is due to different tastes or location each consumer has a different preference for the brands sold in the market There could be two interpretations of location for the environment modeled in this section Location can mean the physical location of a particular consumer in which case the consumer observes the prices charged by all stores and then chooses to purchase from the store at which the price plus the transportation cost is minimized Or location can mean a distance between the brand characteristic that a particular consumer views as ideal and the characteristics of the brand actually purchased That is we can view a space say a line interval as measuring the degree of sweetness in a candy bar Consumers located toward the left are those who prefer lowsugar bars whereas those who are located toward the right prefer highsugar bars In this case the distance between a consumer and a firm can measure the consumers disutility from buying a lessthanideal brand This disutility is equivalent to the transportation cost in the previous interpretation We analyze only horizontally differentiated products That is we analyze brands that are not uniformly utility ranked by all consumers More precisely horizontally differentiated brands are ones that if sold for identical prices elicit from different consumers choices of different brands called ideal brands The analysis of vertically differentiated brands that is brands that are uniformly ranked by all consumers is postponed to section 122 where we discuss product differentiation with respect to quality see more on these issues in Beath and Katsoulacos 1991 and Anderson Palma and Thisse 1992 for a survey see Gabszewicz and Thisse 1992 731 The linear approach Hotelling 1929 considers consumers who reside on a linear street with a length of L 0 Suppose that the consumers are uniformly distributed on this interval so at each point lies a single consumer Hence the total number of consumers in the economy is L Each consumer is indexed by so x is just a name of a consumer located at point x from the origin Price game with fixed location Suppose that there are two firms selling a product that is identical in all respects except one characteristic which is the location where it is sold That is Figure 77 shows that firm A is located a units of distance from Page 150 point 0 Firm B is located to the right of firm A b units of distance from point L Assume that production is costless Figure 77 Hotellings linear city with two firms Each consumer buys one unit of the product To go to a store a consumer has to pay transportation cost of τ per unit of distance Thus a consumer located at some point z has to pay transportation cost of τx a for shopping at firm A or τx L b for shopping at firm B The reader should note that distance here can have a different interpretation We can think of a candy bar that can be produced with different degrees of sweetness Thus if we let x measure the percentage of sugar put into a candy bar firm B produces a sweeter candy than firm A A consumer located at x desires x degree of sweetness more than any other degree of sweetness However the firms offer most consumers degrees of sweetness that differ from the most preferred one With this interpretation the equivalent of transportation costs is the monetary equivalent loss to a consumer who desires x degree of sweetness but instead has to purchase a candy bar with a different degree of sweetness Let us define the utility function of a consumer located at point x by Let denote the consumer who is indifferent to whether he or she purchases from A or B Formally if then Hence which is the demand function faced by firm A The demand function faced by firm B is Page 151 We now look for a BertrandNash equilibrium in price strategies That is Firm A takes pB as given and chooses pA to The firstorder condition is given by Firm B takes pA as given and chooses pB to The firstorder condition is given by Hence the equilibrium prices are given by The equilibrium market share of firm A is given by Note that if a b then the market is equally divided between the two firms The profit of firm A is given by which shows that the profit of each brandproducing firm increases with the distance between the firms This is not surprising in view of the fact that Propositions 71 and 72 showed firms reach higher profit levels when the brands they produce are more differentiated In fact Hotelling 1929 50 states These particular merchants would do well instead of organizing improvement clubs and booster associations to better the roads to make transportation as difficult as possible Page 152 We leave it to the reader to determine whether such a behavior is observed or unobserved The above calculations were performed under the assumption that an equilibrium where firms charge strictly positive prices always exists The following proposition describes the equilibria and provides precise conditions for existence The proof of the proposition is given in the appendix section 75 Proposition 76 1 If both firms are located at the same point a b L meaning that the products are homogeneous then pA PB 0 is a unique equilibrium 2 A unique equilibrium exists and is described by 721 and 722 if and only if the two firms are not too close to each other formally if and only if the unique equilibrium is given by 721 722 and 723 When the two firms are located too closely they start undercutting each others prices resulting in a process of price cuts that does not converge to an equilibrium Proposition 76 shows that in order for an equilibrium to exist the firms cannot be too closely located Location and price game So far we have assumed that the location of the firms is fixed say by the regulating license issuing authority It would be nice to have a theory under which firms can choose price and location Unfortunately we now show that there is no solution for this twodimensional strategy game To show that we ask what would firm A do if given the price and location of its opponent it would be allowed to relocate To answer that 723 implies that meaning that for any locations a and b firm A could increase its profit by moving toward firm B obviously to gain a higher market share This case where firms tend to move toward the center is called in the literature the principle of minimum differentiation since by moving toward the center the firms produce lessdifferentiated products However Page 153 Proposition 76 shows that if firm A gets too close to firm B an equilibrium will not exist Also if firm A locates at the same point where firm B locates its profit will drop to zero implying that it is better off to move back to the left Hence Proposition 77 In the Hotelling linearcity game there is no equilibrium for the game where firms use both prices and location as strategies Quadratic transportation cost Proposition 721 shows that even when the location is fixed the linearlocation model does not have an equilibrium in a price game when the firms are too close to each other We also showed that there is no equilibrium in a game when firms choose both prices and location However it is important to observe that so far we have assumed linear transportation costs The existence problem can be solved if we assume quadratic transportation costs That is let 717 be written as To have even more fun using the quadratictransportationcost setup we can formulate a two period game in which firms decide where to locate in the first period and set prices in the second period Since we look for a SPE Definition 210 the reader who is eager to solve this game should follow the following steps Second period 1 For given location parameters a and b find the NashBertrand equilibrium prices following the same steps we used in order to derive 721 2 Substitute the equilibrium prices into the profit functions 718 and 720 to obtain the firms profits as functions of the location parameters a and b First period Maximize the firms profit functions which you calculated for the second period with respect to a for firm A and with respect to b for firm B Prove that for a given b meaning that firm A would choose a 0 Similarly show that firm B would locate at point L This exercise shows that when there are quadratic transportation costs firms will choose maximum differentiation This result is consistent with Propositions 71 and 72 showing that profits increase with differentiation Page 154 732 The circular approach Proposition 77 shows that an equilibrium in games in which ruins jointly decide on prices and location does not exist in the Hotelling model One way to solve this problem is to let the city be the unitcircumference circle where the consumers are uniformly distributed on the circumference As with the Hotelling model this location model can also be given an interpretation for describing differentiated products that differs from the physicallocation interpretation Consider for example airline bus and train firms which can provide a roundtheclock service If we treat the circle as twentyfour hours each brand can be interpreted as the time where an airline firm schedules a departure Firms This model does not explicitly model how firms choose where to locate However it assumes a monopolisticcompetition market structure in which the number of firms N is endogenously determined All infinitely many potential firms have the same technology Denoting by F the fixed cost by c the marginal cost and by qi and πiqi the output and profit levels of the firmproducing brand i we assume that Consumers Consumers are uniformly distributed on the unit circle We denote by τ the consumers transportation cost per unit of distance Each consumer buys one unit of the brand that minimizes the sum of the price and transportation cost Assuming that the N firms are located at an equal distance from one another yields that the distance between any two firms is 1N Figure 78 illustrates the position of firm 1 relative to the positions of firm 2 and firm N Then assuming that firms 2 and N charge a uniform price p the consumer who is indifferent to whether he or she buys from firm 1 or firm 2 similarly firm N is located at determined by Hence Since firm 1 has customers on its left and on its right the demand function facing firm 1 is Page 155 Figure 78 The position of firms on the unit circle Defining and solving for the monopolisticcompetition equilibrium Let us begin with a definition Definition 74 The triplet Npq is an equilibrium if 1 Firms Each firm behaves as a monopoly on its brand that is given the demand for brand i 727 and given that all other firms charge pj p each firm i chooses p to 2 Free entry Free entry of firms brands will result in zero profits πiq 0 for all i 1 2 N The firstorder condition for firm is maximization problem is Therefore in a symmetric equilibrium Pi p c τN To find the equilibrium number of brands N we set Hence Page 156 Welfare We would like to investigate whether the free market produces a larger or a smaller variety than the optimal variety level Before defining the economys welfare function we calculate the economys aggregate transportation costs denoted by T Figure 78 shows that in equilibrium all consumers purchasing from firm 1 say are located between 0 and 12N units of distance from the firm on each side Since there are 2N such intervals the economys total transportation cost is given by An alternative way to find the aggregate transportation cost without using integration is to look at the cost of the average consumer who is located half way between and a firm That is the average consumer has to travel 14N which yields 729 We define the economys loss function LF τ N as the sum of the fixed cost paid by the producing firms and the economys aggregate transportation cost Formally the Social Pioneer chooses the optimal number of brands N to The firstorder condition is Hence Therefore in a freeentry location model too many brands are produced Notice that there is a welfare tradeoff between the economies of scale and the aggregate transportation cost That is a small number of brands is associated with lower average production costs but higher aggregate transportation costs because of fewer firms A large number of brands means a lower scale of production higher average cost but with a lower aggregate transportation cost Equation 731 shows that it is possible to raise the economys welfare by reducing the number of brands 733 Sequential entry to the linear city So far we have not discussed any model in which firms strategically choose where to locate In subsection 731 we have shown that the basic linearstreet model does not have an equilibrium where firms choose both prices and location Page 157 In this subsection we discuss an example set forth by Prescott and Visscher 1977 in which prices are fixed at a uniform level set by the regulator For example in many countries prices of milk bread and basic cheese products are regulated by the government Thus the only choice variable left to firms is where to locate what characteristics degree of sweetness in our examplethe product should have Consider the unit interval street where there are three firms entering sequentially In this three period model firm 1 enters in period 1 firm 2 in period 2 and firm 3 in period 3 We look for a SPE see Definition 210 in location strategies where each firm maximizes its market share We denote by the location strategy chosen by firm i in period i i 1 2 3 Let denote a very small number representing the smallest possible measurable unit of distance Solving the entire threeperiod game is rather complicated Instead we shall assume that firm 1 has already moved and located itself at the point x1 14 Figure 79 illustrates the location of firm 1 Figure 79 Sequentiallocation game The thirdperiod subgame Firm 3 decides on its location x3 after firm I and firm 2 are already located There are three possible locations of firm 2 corresponding to the three upper parts of Figure 79 Page 158 In this case firm 3 would locate at Here while In this case firm 3 would locate to the right of firm 2 at Here while That is firm 2 shares the x1 x2 interval with firm 1 In this case firm 3 would locate between firm 1 and firm 2 at any point x1x3x2 With no loss of generality assume that Here and The secondperiod subgame Firm 2 knows that in the third period the location decision of firm 3 will be influenced by its own choice of location Thus firm 2 calculates the bestresponse function of firm 3 which we calculated above Hence firm 2 takes the decision rule of firm 3 as given and chooses x2 that would maximize its profit Clearly firm 2 will not locate at since this location yields a maximum profit of it will collect a higher profit by locating elsewhere as described below If firm 2 locates at we have shown that and However if firm 2 locates at we have shown that which is maximized at Located at the profit of firm 2 is In summary the SPE is reached where The bottom part of Figure 79 illustrates the location of the firms in a SPE 734 Calculusfree location model In this subsection we develop a calculusfree version of the Hotelling linearcity model analyzed in subsection 731 Consider a city where consumers and producers are located only at the citys edges Suppose that the city consists of N0 consumers located at point x 0 and NL consumers located at the point x L There are two firms firm A is located also at x 0 and firm B is located at Page 159 Figure 710 Discretelocation model x L Assume that production is costless Figure 710 illustrates the location of firms and consumers in this city Each consumer buys one unit either from the firm located where the consumer is or from the firm located on the other side of town Shopping nearby does not involve transportation cost whereas shopping on the other side of town involves paying a fixed transportation cost of Let pA denote the price charged by firm A and pB the price charged by firm B Thus we assume that the utility of the consumer located at point x 0 is given by Similarly the utility of the consumer located at point x L is given by Let nA denote the number of consumers buying from firm A and nB denote the number of consumers buying from firm B Then 733 and 734 imply that Nonexistence of a NashBertrand equilibrium A NashBertrand equilibrium is the nonnegative pair such that for a given firm A chooses to maximize and for a given firm B chooses to maximize where nA and nB are given in 735 Page 160 Proposition 78 There does not exist a NashBertrand equilibrium in prices for the discrete version of Hotellings location model Proof By way of contradiction suppose that constitute a Nash equilibrium Then there are three cases i ii and iii i With no loss of generality suppose that Then 735 implies that and hence However firm A can deviate and increase its profit by reducing its price to and by having ñA N0 thereby earning a profit of A contradiction ii With no loss of generality suppose that Then firm A can deviate and increase its profit by slightly increasing its price to satisfying and maintaining a profit level of A contradiction iii With no loss of generality suppose that Then Hence as firm A did in case ii firm B can increase its profit by slightly raising A contradiction Undercutproof equilibrium Since a Nash equilibrium in prices for the discretelocation model does not exist in this subsection we define motivate and solve for the undercutproof equilibrium In an undercutproof equilibrium each firm chooses the highest possible price subject to the constraint that the price is sufficiently low so that the rival firm would not find it profitable to set a sufficiently lower price in order to grab the entire market That is in an undercutproof equilibrium firms set prices at the levels that ensure that competing firms would not find it profitable to completely undercut these prices Thus unlike behavior in a NashBertrand environment where each firm assumes that the rival firm does not alter its price in an undercutproof equilibrium environment firms assume that rival firms are ready to reduce their prices whenever undercutting prices and grabbing their rivals market are profitable to them This behavior is reasonable for firms competing in differentiated products Definition 75 An undercutproof equilibrium for this economy is nonnegative and such that 1 For given and firm A chooses the highest price subject to Page 161 2 For given and firm B chooses the highest price subject to 3 The distribution of consumers between the firms is determined in 735 Part 1 of Definition 75 states that in an undercutproof equilibrium firm A sets the highest price under the constraint that the price is sufficiently low to prevent firm B from undercutting and grabbing the entire market More precisely firm A sets sufficiently low so that Bs equilibrium profit level exceeds Bs profit level when it undercuts by setting and grabbing the entire market nB N0 NL Part 2 is similar to part 1 but describes how firm B sets its price We proceed with solving for the equilibrium prices Proposition 79 There exists a unique undercutproof equilibrium for the discretelocation problem given by and Proof First note that by setting each firm can secure a strictly positive market share without being undercut Hence in an undercutproof equilibrium both firms maintain a strictly positive market share From 735 we have it that and Substituting and into the two constraints in Definition 75 and then verifying 735 yields the unique undercutproof equilibrium Figure 711 illustrates how the undercutproof equilibrium is determined The left side of Figure 711 shows how firm A is constrained in setting pA to fall into the region where firm B would not benefit from undercutting compare with part 1 in Definition 75 The center of Figure 711 shows how firm B is constrained in setting pB to fall into the region where firm A would not benefit from undercutting compare with part 2 in Definition 75 The right side of Figure 711 illustrates the region where neither firm finds it profitable to undercut the rival firm and the undercutproof equilibrium prices It should be emphasized that the curves drawn in Figure 711 are not best response reaction functions The curves simply divide the regions into prices that make undercutting profitable or unprofitable for one firm Page 162 Figure 711 Undercutproof equilibrium for the discretelocation model Properties of the undercutproof equilibrium Clearly prices rise with transportation costs and monotonically decline to zero as transportation costs approach zero reflecting a situation in which the products become homogeneous More interestingly Hence if and only if Thus in an undercutproof equilibrium the firm selling to the larger number of consumers charges a lower price This lower price is needed to secure the firm from being totally undercut Finally under symmetric distribution of consumers N0 NL the equilibrium prices are given by That is each firm can mark up its price to twice the level of the transportation cost without being undercut 74 Appendix Inverting Demand Systems The demand system 71 can be written as Define Δ to be the determinant of Page 163 Then using Cramers Law we have it that This establishes equation 72 75 Appendix Existence of an Equilibrium in the Linear City We now prove Proposition 76 1 When a b 1 the products are homogeneous so the undercutting procedure described in section 63 applies 2 For the general proof see dAspremont Gabszewicz and Thisse 1979 Here we illustrate the argument made in their proof for the simple case where firms are located at equal distances along the edges That is assume that a b a L2 Then we are left to show that the equilibrium exists if and only if or if and only if When a b the distance between the two firms is L 2a Also if equilibrium exists 721 is now given by pA pB τL The profit level of firm A as a function of its own price pA and a given Bs price for the case of a b is drawn in Figure 712 Figure 712 Existence of equilibrium in the linear city The profit of firm A for a given Page 164 Figure 712 has three regions Region I Here pA τL τL 2a In this case pA is very low so that even the consumer located at the same point where firm B is located would purchase from firm A Thus firm A has the entire market and its profit is given by υA pAL Region II Here both firms sell a strictly positive amount so the profit of firm A as a function of pA is given in equation 718 Substituting the equilibrium pB τL into 718 yields which is drawn in Region II of Figure 712 Maximizing 738 with respect to pA yields πA τL22 which corresponds to the peak drawn in Figure 712 Region III Here pA is high so all consumers purchase from firm B This is the polar case of Region I Now for a given pB τL Figure 712 shows that πA has two local maxima In one it has the entire market share pA τL τL 2a e whereas in the other it shares the market with firm B pA τL For 721 to constitute the equilibrium prices we must have it that in equilibrium the globally profitmaximizing price for firm A would lie in Region H and not Region I Or that for the equilibrium pB τL implying that 76 Exercises 1 Suppose that there are only two firms selling coffee called firms 1 and 2 Let αi denote the advertising level of firm i i 1 2 Assume that the profits of the firms are affected by the advertising levels taken by the firms Formally assume that Answer the following questions a Calculate and draw the bestresponse function of each firm That is for any given advertising level of firm j find the profitmaximizing advertising level of firm i b Infer whether the strategies are strategically complements or strategically substitutes see Definition 72 Page 165 c Find the Nash equilibrium advertising levels Also calculate the firms Nash equilibrium profit levels 2 Consider the Hotelling linearcity model analyzed in Subsection 731 Suppose that in the linear city there is only one restaurant located at the center of the street With a length of 1 km Assume that the restaurants cost is zero Consumers are uniformly distributed on the street which is the interval 0 1 where at each point on the interval lives one consumer Suppose that the transportation cost for each consumers is 1 for each unit of distance each kilometer of travel The utility of a consumer who lives a units of distance from the restaurant is given by where p is the price of a meal and B is a constant However if the consumer does not eat at the restaurant her utility is U 0 Answer the following questions a Suppose that the parameter B satisfies 0 B 1 Find the number of consumers eating at this restaurant Calculate the monopoly restaurants price and profit levels b Answer the previous question assuming that B 1 3 University Road is best described as the interval 0 1 Two fastfood restaurants serving identical food are located at the edges of the road so that restaurant 1 is located on the most left hand side and restaurant 2 is located on the most righthand side of the road Consumers are uniformly distributed on the interval 0 1 where at each point on the interval lives one consumer Each consumer buys one meal from the restaurant in which the price plus the transportation cost is the lowest In University Road the wind blows from right to left hence the transportation cost for a consumer who travels to the right is R per unit of distance and only 1 per unit of distance for a consumer who travels to the left Answer the following questions a Let pi denote the price of a meal at restaurant i i 1 2 Assume that p1 and p2 are given and satisfy Denote by the location of the consumer who is indifferent to whether he or she eats at restaurant 1 or restaurant 2 and calculate as a function of p1 p2 and R b Suppose that the given prices satisfy p1 p2 What is the minimal value of the parameter R such that all consumers will go to eat only at restaurant 1 4 Consider the Hotelling model with quadratic transportation cost described in equation 724 and assume that both firms are located at the same distances from the edges of the unit interval ie in Figure 77 Page 166 a Assuming that firms produce the product with zero cost calculate the symmetric Nash equilibrium in prices b Assuming that firm A is allowed to make a small adjustment in its location before both firms choose their prices would firm A move inward or outward Prove your answer 77 References Anderson S A Palma and J Thisse 1992 Discrete Choice Theory of Product Differentiation Cambridge Mass MIT Press Beath J and Y Katsoulacos 1991 The Economic Theory of Product Differentiation Cambridge Cambridge University Press Bulow J J Geanakoplos and P Klemperer 1985 Multimarket Oligopoly Strategic Substitutes and Complements Journal of Political Economy 93 488511 Chamberlin E 1933 The Theory of Monopolistic Competition Cambridge Mass Harvard University Press Cheng L 1985 Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146152 Chou C and O Shy 1991 Intraindustry Trade and the Variety of Home Products Canadian Journal of Economics 24 405416 dAspremont C J Gabszewicz and J Thisse 1979 On Hotellings Stability in Competition Econometrica 17 11451151 Dixit A 1979 A Model of Duopoly Suggesting a Theory of Entry Barriers Bell Journal of Economics 10 2032 Dixit A and J Stiglitz 1977 Monopolistic Competition and Optimum Product Diversity American Economic Review 67 297308 Eaton B C and R Lipsey 1989 Product Differentiation In Handbook of Industrial Organization edited by R Schmalensee and R Willig Amsterdam NorthHolland Gabszewicz J and J Thisse 1992 Location In Handbook of Game Theory edited by R Aumann and S Hart Amsterdam NorthHolland Gros D 1987 A Note on the Optimal Tariff Retaliation and the Welfare Loss from Tariff Wars in a Model with IntraIndustry Trade Journal of International Economics 23 457367 Hotelling H 1929 Stability in Competition Economic Journal 39 4157 Helpman E and P Krugman 1985 Market Structure and Foreign Trade Cambridge Mass MIT Press Lancaster K 1979 Variety Equity and Efficiency New York Columbia University Press Prescott E and M Visscher 1977 Sequential Location among Firms with Foresight The Bell Journal of Economics 8 378393 Page 167 Salop S 1979 Monopolistic Competition with Outside Goods Bell Journal of Economics 10 141156 Sattinger M 1984 Value of an Additional Firm in Monopolistic Competition Review of Economic Studies 51 321332 Scherer F M 1979 The Welfare Economics of Product Variety An Application to the ReadyTo Eat Cereals Industry Journal of Industrial Economics 28 113133 Schmalensee R 1978 Entry Deterrence in the ReadyToEat Breakfast Cereal Industry Bell Journal of Economics 9 305327 Singh N and X Vives 1984 Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546554 Vives X 1985 Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166175 Page 169 Chapter 8 Concentration Mergers and Entry Barriers A prime reason for studying industrial organization is for understanding why concentration is observed very often Common statement As we discussed in the introduction the study of industrial organization is motivated mainly by the failure of the competitive market structure model analyzed in chapter 4 to explain the commonly observed high concentration of firms in the same industry Therefore in this chapter we attempt to address the following questions 1 Why do firms in some industries make pure profits 2 When oligopolies make pure profits how come entry of new firms does not always occur thereby eliminating all pure profits 3 What can explain mergers among firms in a given industry 4 What is and what should be the regulators attitudes towards concentrated industries More precisely a Should the regulator limit and control mergers among firms in the same industry b Even if mergers do not occur should the regulator attempt to control the degree of concentration in industries Section 81 Concentration Measures discusses and defines methods for measuring the degree of concentration in an industry That is we define indexes for measuring the distribution of market shares across Page 170 firms in a given industry Section 82 Mergers analyzes merger activities among firms and how those activities affect the industrys level of concentration This section investigates the incentives of firms within various industries to merge with other firms in the same industry Section 83 Entry Barriers and section 84 Entry Deterrence provide a wide variety of explanations classified into two related groups for why entry does not always occur despite the fact that existing firms in the industry make strictly positive profits By entry barriers we will refer to a long list of conditions that explain why entry does not occur These conditions could be technological such as economies of scale or sunk entry costs legal such as patent protection or exclusive rights given by other firms or regulators or the result of market organization conditions such as distribution channels marketing networks or consumer loyalty and goodwill All these conditions are discussed in section 83 By entry deterrence we will refer to strategic actions taken by incumbent firms when faced with a threat of actual entry into their industry By strategic actions we mean actions that the incumbent firm would not find profitable to take in the absence of entry threats Analyzing all possible such actions is the subject of section 84 The distinction between entrybarrier arguments and entrydeterrence arguments is not without troubles for several reasons In many cases it is hard to find whether the conditions leading to no entry are external to the firms or are created by the incumbent firms This in most cases makes antitrust litigation against monopoly firms very difficult because the monopoly firm can claim that the conditions that prevent entry are external to the firms Furthermore some of the conditions preventing entry can be augmented by the incumbents behavior More precisely we will show that the existence of sunk irreversible costs may be sufficient to sustain one monopoly firm in the industry Now note that some sunk costs are external to the firms such as entry taxes paid to the local authorities initial market surveys required by the investors and so on However there are many sunk costs that are firm dependent For example the incumbent firm may spend on RD to improve its product for the purpose of forcing RD costs on the potential entrant In addition the incumbent may spend large sums of money on advertising for the purpose of forcing advertising sunk cost on the potential entrant In most of our analysis sunk cost is either explicitly assumed or implicitly assumed to prevail as a consequence of having firms committing to certain capacityoutput levels Section 85 Contestable Markets introduces a contestable market structure which describes the behavior of an incumbent firm when potential entrants can enter without having to bear any sunk cost generally called hit andrun entry Finally an appendix section 86 provides an overview on how the Page 171 Department of Justice and the Federal Trade Commission decide whether to challenge a merger and the corresponding operating guidelines Appendix section 87 discusses the legal approach to entry deterrence behavior 81 Concentration Measures So far our discussion of industry concentration regarded concentrated industry as one where there are few firms and each firm maintains a high market share In this section we eliminate the vagueness behind the concept of concentration and propose precise measures of concentration There are two reasons why there is a need for these precise measures First to be able to compare concentration among different industries in the same or different countries The compared industries need not share anything in common but a proper concentration measure should be able to compare concentration despite the fact that different industries have different numbers of firms and different distributions of market shares Second in case the regulating authority would like to intervene and to prevent a change in concentration of a certain industry the regulator must specify a general measure by which it decides that a certain industry is concentrated These measures can then be used by the legal system that arbitrates conflicts between the firms and the regulator about mergers What is a concentrated industry Clearly the most concentrated industry is a monopoly which sells 100 of the industrys output When the number of firms is greater than one there are two factors that influence concentration a the number of firms in the industry and b the distribution of output among the firms in the industry Thus a measure of concentration should be sensitive to both the distribution of the industrys output across firms as well as the number of firms in the industry Let N be the number of firms in the industry let Q denote the aggregate industryoutput level aggregate amount sold to consumers and let qi denote the output of firm i i 1 2 N Thus Obviously there may be a problem with this summation if the industry is composed of firms producing differentiated products In other words can we add red cars and purple cars What about adding large cars and small cars We ignore these aggregation problems which come up in almost every empirical work in industrial organization and international trade Let denote the percentage of the industrys total output sold by firm i We call si the mariner share of firm i Observe Table 81 Measures for industrys concentration si in percentage Page 172 that and that In what follows we discuss two commonly used measures of concentration which among other indicators are used by the US Federal Trade Commission for determining whether to approve a merger For more measures and their interpretation see discussions in Jacquemin 1987 and Tirole 1988 Chapter 5 811 The fourfirm concentration ratio The fourfirm concentration ratio was used for merger guidelines see an appendix section 862 purposes from 1968 to around 1982 It simply sums up the market shares of the four largest firms in the industry Let us order all the firms in the industry rename them so that firm 1 would have the largest market share firm 2 the second largest and so on That is We define the fourfirm concentration ratio by Table 81 demonstrates the value of I4 for four imaginary industries share s1 s2 s3 s4 s5 s6 s8 s9 s10 I4 IHH Industry 1 60 10 5 5 5 0 80 3850 Industry 2 20 20 20 20 0 0 80 2000 Industry 3 0 0 0 100 3333 Industry 4 49 49 025 025 025 025 985 4802 You probably notice that there is something unsatisfactory about the fourfirm concentration ratio In industry 1 firm 1 has 60 of the market Industry 2 has five firms all have equal market shares of 20 However the fourfirm concentration ratio yields I4 80 for both industries We conclude that since the fourfirm measure is linear it does not differentiate between different firm sizes as long as the largest four firms maintain most of the market shares Comparing industries 3 Page 173 and 4 demonstrates the same problem where an industry equally shared by three firms is measured to be more concentrated than an industry dominated by only two firms 812 The HerfindahlHirshman index The HerfindahlHirshman index denoted by IHH is a convex function of firms market shares hence it is sensitive to unequal market shares We define this measure to be the sum of the squares of the firms market shares Formally Table 81 shows that the IHH for industry 1 is almost twice the IHH for industry 2 This follows from the fact that squaring the market shares of the large firms increases this index to a large value for industries with significantly unequal market shares Comparing industries 3 and 4 shows that while the I4 measure indicates that industry 3 is more concentrated than industry 4 the IHH measure indicates that industry 4 is more concentrated than industry 3 For this reason the IHH is found to be the preferred concentration measure for regulation purposes 82 Mergers The terms mergers takeovers acquisitions and integration describe a situation where independently owned firms join under the same ownership We will use the term merger to refer to any type of joining ownership and disregard the question of whether the merger is initiated by both firms or whether one firm was taken over by another Instead we investigate the gains and incentives to merge and the consequences of mergers for the subsequent performance and productivity of the firms involved for consumers welfare and for social welfare The Federal Trade Commission classifies mergers into three general categories Horizontal merger This occurs when firms in the same industry producing identical or similar products and selling in the same geographical market merge Vertical merger This occurs when a firm producing an intermediate good or a factor of production merges with a firm producing the final good that uses this intermediate good or when two companies who have a potential buyerseller relationship prior to a merger merge Page 174 Conglomerate merger This occurs when firms producing less related products merge under the same ownership More precisely conglomerate mergers are classified into three subclasses Product extension The acquiring and acquired firms are functionally related in production or distribution Market extension The firms produce the same products but sell them in different geographic markets Other conglomerate The firms are essentially unrelated in the products they produce and distribute Ravenscraft and Scherer 1987 provide a comprehensive study of merger activities in the United States and report four great merger waves that have marked American industrial history one peaking in 1901 a milder one during the late 1920s a third with its peak in 1968 and the most recent one a resurgence in the early 1980s Looking at the types of mergers we note that the data show a significant decline in horizontal and vertical activity and a rise in pure conglomerate mergers from the 1960s The merger wave of the turn of the century was preponderantly horizontal The wave of the 1920s saw extensive activity in the publicutility sector in vertical and productline extension and in horizontal mergers that created oligopolies rather than monopolies The wave of the 1960s was preponderantly conglomerate reflecting a much more stringent antitrust policy against horizontal mergers Why do mergers occur First a merger may reduce market competition between the merged firms and other firms in the industry thereby increasing the profit of the merged firms However note that section 88 exercise 2 demonstrates in a Cournot market structure that when there are more than two firms in the industry the aggregate profit of the merged firms can be lower than the profit of the two firms separately before the merger occurs Second if the merger involves merging capital assets and other fixed factors of production then the merged firms would be able to increase their size possibly reduce cost and thereby increase their market share hence profit Third mergers and takeovers occur when there is a disparity of valuation judgments given uncertainty about future business conditions the buyer is for some reason more optimistic about the firms future than the seller or the buyer believes it can run the acquired entity more profitably as a part of this organization than the seller could by remaining independent Fourth those who control the acquiring entity seek the prestige and monetary rewards associated with managing a large corporate empire whether or not the consolidation adds to the profits Page 175 821 Horizontal merger In subsection 613 we saw some theoretical basis for the presumption that under a Cournot market structure a decrease in the number of firms in an industry via say a merger reduces social welfare That is we have shown that under a Cournot market structure in the case of identical firms with no fixed costs an increase in the number of firms increases the sum of consumer surplus and producers profits despite the fact that profits decline However there is still a question of whether a regulator should refuse to permit a merger to take place only on the basis of the associated sharp increase in concentration The answer to this question is no That is in what follows we construct an example where a merger of a highcost firm with a lowcost firm increases overall welfare despite the increase in concentration for a comprehensive analysis of mergers under the Cournot market structure for the case of n firms see Salant Switzer and Reynolds 1983 Consider the Cournot duopoly case that of two firms producing a homogeneous product analyzed in subsection 611 on page 98 Let the unit costs be c1 1 and c2 4 and the demand be p 10 Q Equations 65 66 and 67 imply that under the Cournot duopoly market structure pc 10 4 1 5 Hence in view of 33 see subsection 323 the consumer surplus is Hence in view of 613 Now allow a merger between the two firms The new firm is a multiplant monopoly and as shown in section 54 the newly merged firm would shut down plant number 2 Hence the merged firm solves a simple singleplant monopoly problem analyzed in section 51 yielding an output level of Qm 45 and pm 10 45 55 hence πm 55 145 814 Also CS45 ½10 552 818 Altogether Wm CS45 πm 30375 Comparing the premerger concentration level with the postmerger monopoly yields that Observing that Wm Wc we can state the following Proposition 81 Under a Cournot market structure a merger among firms leading to an increase in concentration does not necessarily imply an overall welfare reduction The intuition behind Proposition 81 is that when firms have different production costs there exists a tradeoff between production efficiency Page 176 and the degree of monopolization In other words a merger between a highcost and a lowcost firm increases production efficiency since it eliminates the highcost producer However the increase in concentration increases the market price and therefore reduces consumer welfare Now when the difference in production costs between the two firms is significant the increase in production efficiency effect dominates the reduction in consumer welfare In view of the merger guidelines described in subsection 862 such a merger will not be approved despite this examples demonstration that the merger would improve overall welfare However the reader is advised not to take this example too seriously for the following reason It is possible that our methodology is wrong in the sense that we are making welfare judgments based on the Cournot market structure Had the firms played Bertrand the inefficient firm firm 2 would not be producing in the duopoly case In summary conclusions about welfare that are based only on the Cournot market structure should be checked to determine whether they also hold under different market structures Otherwise such a welfare analysis is not robust The analysis of this subsection has a major shortcoming in that it is done without accounting for firms size and therefore for the effects of changes in size associated with every merger That is under a Cournot market structure when two firms with the same unit costs merge their actual size merges into a single firm Davidson and Deneckere 1984 develop a model that overcomes this shortcoming by introducing capacity to the analysis In their model when two firms with invested capacity merge they merge with their entire stock of capacity so the joint firm maintains a larger capacity level than each individual firm 822 Vertical merger A vertical merger is defined as a merger between a supplier producer of an intermediate good and a producer of a final good who uses this intermediate good as a factor of production The common terminology used to describe these firms is to call the intermediategood suppliers as upstream firms and the finalgood producers as downstream firms Figure 81 illustrates an industry structure in which there are two upstream firms selling an input to two downstream firms In Figure 81 the two input suppliers denoted by A and B sell identical inputs to both downstream firms denoted by 1 and 2 The lefthand side of Figure 81 shows the initial situation in which all firms are disjoint The righthand side illustrates the case in which the upstream firm A merges with downstream firm 1 We denote the merged firm by A1 There are several ways in which competition in the upstream and downstream markets could Page 177 Figure 81 Upstream factor suppliers and downstream producers be modeled See for example Ordover Saloner and Salop 1990 Perry 1989 Salinger 1988 and Tirole 1988 Chap 4 Clearly if both the upstream and the downstream markets are characterized by a Bertrand price competition then it is easy to show that profits of all firms are identically zero before and after vertical integration occurs In order to solve this modeling problem we could assume that the downstream firms produce differentiated products such as the Hotelling spatial competition analyzed in subsection 731 so that firms would make positive profits Instead we take an approach yielding similar results by assuming that the upstream market is characterized by a Bertrand price competition section 63 whereas the downstream market is characterized by a Cournot quantity competition section 61 Downstream competition We assume that the demand for the good marketed in the downstream market is given by the linear demand p α q1 q2 where α 0 and q1 and q2 are the output levels sold by downstream firms 1 and 2 Let the technology be such that one unit of input produces one unit of output and denote by c1 and c2 the price of the input paid by firms 1 and 2 respectively Hence the firms unit costs are given by c1 and c2 respectively In section 61 we showed that under this demand and cost structure a Cournot quantity competition yields the output and profit for each firm i given by Hence the aggregate downstream production and price levels are Page 178 Upstream competition before the merger The upstream firms A and B sell the intermediate product to the downstream firms 1 and 2 Since the two upstream firms engage in a Bertrand price competition prices fall to their unit production cost which is assumed to be zero Hence c1 c2 0 so the downstream firms have zero production costs Thus substituting into 84 yields Upstream and downstream merge Suppose now that upstream firm A merges with downstream firm 1 We denote the merged firm by A1 Hence the input cost of the merged firm A1 is zero We assume that the merged firm A1 does not sell the intermediate good to firm 2 therefore the upstream firm B is now a monopoly in the factor market and maximizes its profit by choosing the price for its intermediate product c2 that equals the cost of production of downstream firm 2 Thus the profit of upstream firm B is its price c2 times the output level of downstream firm 2 given in 84 Formally the upstream firm B chooses c2 that solves The firstorder condition yields 0 α 4c2 c1 yielding that c2 α4 Clearly the secondorder condition is satisfied so substituting c1 0 and c2 α4 into 84 and 85 yields Hence the profit of the two downstream firms is given by Equation 88 yields the following proposition Proposition 82 A merger between an upstream and a downstream firm increases the output level of the merged firm and reduces the output level of the downstream firm that does not merge Proposition 82 is rather intuitive The downstream firm that does not merge faces an increase in its input cost resulting from having to buy its input from a single monopoly firm B Hence the increase in firm 2s Page 179 production cost and the reduction in firm 1s production cost would increase the output of firm 1 and reduce the output of firm 2 We wish to investigate whether this vertical merger is profitable to the vertically merging firms To see that we need to compare the sum of profits of firms A and 1 prior to the merger to the profit of the merged firm A1 However prior to the merger firm A made zero profit hence prior to merger their joint profit was π1 α29 Comparing this sum to πA1 in 89 implies that Proposition 83 1 The combined profit of the merging upstream and downstream firms increase after they merge 2 A merger between the upstream and the downstream firms will not foreclose the market of the disjoint downstream firm but will only reduce its profit Proposition 83 is important since it is often argued that vertical mergers lead to a foreclosure of the disjoint downstream firms which in our example means that firm B or firm 1 or both would go out of business Note that this cannot happen in the present model since the upstream firm B will reduce the input price to prevent firm 2 from leaving the market firm B sells only to firm 2 after the merger Since vertical integration does not necessarily imply foreclosure the FTC seems to be more forgiving to vertical mergers than to horizontal mergers Moreover many economists believe that vertical integration should be viewed as an increase in efficiency since most firms carry on several stages of production under a single plant anyway with or without vertical integration Thus a firm is by definition a vertically merged entity and is believed to be an efficient form of organization Finally the sum of the profits of the disjoint upstream firm B and downstream firm 2 is given by which is the sum of profits of firm B and 2 prior to the merger between firm A and firm 1 Thus despite the fact that the profit of the nonmerging upstream firm B increases with the merger of firm A with firm 1 the decline in the profit of the nonmerging finalgoodproducer firm 2 is larger than the increase in πB which is caused by the sharp drop in market share of firm 2 Page 180 823 Horizontal merger among firms producing complementary products It was Cournot who realized that horizontal merger need not increase the equilibrium price level when two firms producing complementary products merge The reader is probably familiar with the definition and examples of complementary products Examples include coffee and milk or sugar audio receivers and speakers video players and cassettes cameras and film computers and monitors computers and software cars and tires transportation and hotel services and more The reader is referred to section 103 for further analyses of the economics of systems that are composed of complementary components In this subsection we analyze an industry where firms produce two complementary products Economides and Salop 1992 provide a more extensive analysis of complementary systems by considering several producers of each product Demand for systems Consider a market for computer systems A computer system is defined as a combination of two complementary products called computers denoted by X and monitors denoted by Y We denote by pX the price of one computer and by pY the price of a monitor Therefore since a system consists of one computer and one monitor the price of a system is given by pS pX pY Let Q denote the quantity of systems purchased by all consumers and assume that the aggregate consumer demand is given by We denote by x the amount of computers sold to consumers and by y the amount of monitors sold Since the two components are perfect complements x y Q Independently owned producing firms Suppose that computers and monitors are produced by different firms whose strategic variables are prices and suppose that production of either product is costless Consider the problem solved by the computer firm Xproducer For a given pY firm X chooses pX that solves The firstorder condition yields Clearly the secondorder condition is satisfied Hence firm Xs pricebestresponse Page 181 function to Ys price is px α py2 Similarly we can show that Ys price best response with respect to Xs price is py α px2 Altogether when the complementary components are produced by independent firms their prices quantities and firms profit levels are given by Monopoly producing all components Now suppose that firms X and Y merge under a single ownership Thus computers are now sold as systems composed of a single monitor bundied with a single computer Therefore the monopoly systems producer chooses a system price ps that solves yielding a firstorder condition given by Clearly the secondorder condition is satisfied Hence the price of a system under monopoly and the monopolys profit are given by We conclude the discussion on mergers with the following proposition which follows from the comparison of 813 and 814 Proposition 84 A merger into a single monopoly firm between firms producing complementary products would 1 reduce the price of systems ie 2 increase the number of systems sold ie QM Q and 3 increase the sum of profits of the two firms ie The significance of Proposition 84 is that a merger between two firms producing complementary products can increase social welfare since consumers face lower prices and firms gain a higher profit The intuition behind Proposition 84 is as follows Given that the two components are perfect complements a rise in the price of one component reduces the demand for both components Under price competition among independent componentproducing firms each firm overprices its component since each firm is affected by the reduced demand for its component Page 182 and not the entire system Thus the negative externality on the other firms demand is not internalized However when the firms merge the joint ownership takes into consideration how the demand for both components is affected by an increase in the price of one component and the negative demand externality is internalized We conclude our discussion of merger of firms producing complementary products with two remarks First Sonnenschein 1968 has shown that the Nash equilibrium were firms compete in price and sell perfect complements is isomorphic to the case where firms compete in quantity and sell perfect substitutes One simply has to interchange the roles of price in the network case with the industry quantity in the perfect substitutes case For example Proposition 84 can be reinterpreted as showing that under quantity competition among firms selling perfect substitutes a merger to monopoly would 1 reduce the aggregate quantity produced 2 increase the price and 3 result in strictly larger industry profits Second Gaudet and Salant 1992 show that the merger of firms producing complements and setting prices may be unprofitable if some members of the industry are not parties to the merger Given Sonnenscheins observation their result implies that mergers to less than monopoly may also be unprofitable if firms produce perfect substitutes and engage in Cournot competition a point first noted in Salant Switzer and Reynolds 1983 83 Entry Barriers Why do we frequently observe that firms do not enter an industry despite the fact that the existing firms in the industry make above normal profits In this section we investigate the following question If oligopolies make pure profits why does free entry not occur until competition brings down the price so that existing firms will no longer make above normal profits Barriers to entry are considered an important structural characteristic of an industry The competitiveness and the performance of an industry are generally assumed to be strongly influenced by its entry conditions There can be many reasons why entry may not occur The primary explanation for entry barriers is the existence of entry cost Bains pioneering work 1956 specified three sources of entry barriers absolute cost advantages of incumbent firms economies of scale and productdifferentiation advantages of incumbent firms such as reputation and goodwill In addition politicians and all levels of governments may explicitly or implicitly support the existing firms and the existing firms may in return support and contribute to the campaigns of politicians Maintaining such connections seems impossible for new investors Other Page 183 reasons include the learning experience possessed by the existing firms consumers loyalty to brands already consumed and availability of financing banks are less eager to lend to new investors see also Geroski Gilbert and Jacquemin 1990 In this section we briefly discuss entry barriers As we mentioned earlier we regard entry barriers as the conditions that are not controlled by the incumbent firms that explain why entry does not occur Section 84 below will address issues of entry deterrence which we regard as the strategic actions taken by incumbent firms when facing the entry into an industry of potential competitors Subsection 831 demonstrates a technological explanation for entry barriers and shows how the degree of concentration is related to the fixed production costs Subsection 832 demonstrates the role that the existence of sunk costs play in generating the conditions for entry barriers 831 Concentration and fixed costs in a noncompetitive market structure an example Let us demonstrate the relationship between fixed costs and concentration by means of an example Consider the monopolistic competition in the differentiatedproducts environment analyzed in section 72 on page 143 In that environment firms have to bear a fixed cost implying that in equilibrium there will be entry of a finite number of firms More precisely recall from Proposition 75 on page 147 that the number of firms is Nmc L2F where L is the economys resource endowment and F is the fixed cost of each firm L 2F Hence the industry described in Section 72 yields a concentration level given by Consequently in a monopolisticcompetition environment the IHH concentration ratio increases with the fixed cost A similar calculation can be performed in a Cournot market structure where firms have fixed cost and therefore only a finite number of firms would enter For the case of an industry producing a homogeneous product yon Weizsäcker 1980 demonstrates that if production technologies exhibit increasing returns to scale at low output levels Ushaped averagecost functions then the equilibrium number of firms is larger than the social optimum 832 Sunk costs generate entry barriers By sunk costs we mean costs that cannot be reversed or for which the investment associated with paying them cannot be converted to other Page 184 causes or resold in order to recapture part of the investment cost Examples include legal lawyers fees and taxes that an entering firm must bear prior to the actual entry If after paying this cost a firm reverses its decision to enter the firm cannot recover these fees Other forms of sunk costs include market surveys almost always mandated by the investors advertising costs and expenditures on nontransportable nonconvertible plant and equipment such as the site preparation work for any plant Following Stiglitz 1987 we now demonstrate how in a market for a homogeneous product the existence of even small sunk costs can serve as an entry barrier so that entry will not occur even if the incumbent continues to make a monopoly profit There are two firms A and B both capable of producing an identical product with identical constant marginal costs Firm B is the potential entrant If firm B enters it has to sink dollars into the process Firm A is the incumbent monopoly firm earning a profit of where πM denotes the monopolys profit level not including the entry cost it has already sunk in This extensiveform game is illustrated in Figure 82 Figure 82 Sunk costs and entry barriers In the game illustrated in Figure 82 the potential entrant firm B moves first by choosing whether to enter or not In case firm B chooses not to enter it saves the entry cost and therefore earns zero profit In this case firm A remains a monopoly and makes the monopoly profit less than the entry cost it sunk earlier In contrast if firm B enters the firms are assumed to set their prices simultaneously yielding a Bertrand equilibrium see Definition 62 on page 108 where price equals marginal cost In this case both firms make a loss equal to the sunk cost It is straightforward to establish the following proposition Proposition 85 For any level of sunk entry cost satisfyting there exists a unique subgame perfect equilibrium where firm A is a monopoly earning and firm B stays out Page 185 That is in a SPE the entrant foresees that after entry occurs the second stage of the game the incumbent will switch from being a monopoly to being in an aggressive price competition and leading the marginalcost pricing Hence in the first stage the potential entrant will choose not to enter since staying out yields zero profit Proposition 85 is rather disturbing because it means that entry will never occur as long as there are some even infinitesimal sunk costs associated with entry However the reader should notice that Proposition 85 applies only to homogeneous products In fact under these circumstances it is likely that the entrant will engage itself in further investments higher sunk costs in order to develop a differentiated brand in which case price competition need not yield zero or negative profits However Proposition 85 makes a point by stating that even small sunk cost can create all the conditions for entry barriers In fact the incumbent does not need to do anything to deter this entry and simply continues producing the monopoly output level Proposition 85 highlights the role expost competition plays in creating entry barriers What generates the entry barriers even for negligible sunk cost is the intensity of the postentry price competition Had we assumed that the firms play Cournot after entry occurs low sunk cost would not generate entry barriers Assuming Bertrand price competition generates the postentry intense competition that makes entry unprofitable for even low entry costs We conclude this analysis be considering a situation where a firm could receive an amount of ø0 upon exit For example if then we can view ø as the amount of its original expenditure the firm can recover upon exit Figure 83 illustrates the modified game Figure 83 Sunkcost entry barriers with partial cost recovery Page 186 In Figure 83 we added an additional stage enabling the incumbent firm A to exit after firm B makes its choice whether to enter or not In fact firm As exit choice could have been included in the original game Figure 82 however in that game As exit action was clearly dominated by other actions and was therefore ignored We now look for a subgame perfect equilibrium for this game The subgame on the right B does not enter has a unique Nash equilibrium where the incumbent stays in and firm B earns zero profits The subgame on the left starting with the node where the incumbent makes a move has a unique Nash equilibrium where the incumbent exits the industry and collects a profit of it would collect if it stays in In this case the entrant becomes a monopoly Therefore Proposition 86 There exists a unique subgame perfect equilibrium for the game described in Figure 83 where firm B enters and firm A incumbent exits the industry Proposition 86 states that the market for this product will remain dominated by a monopoly market structure despite the fact that entry and exit occur One monopoly replaces another monopoly Hence from the consumers point of view this particular market will be regarded as one that has substantial entry barriers Finally we can further modify the game described in Figure 83 by adding an initial stage in which firm A makes a choice whether to enter the game and become the incumbent firm Clearly if firm A would be able to recover only part of its sunk cost then it would not enter at all and no other firm would ever find it profitable to enter This result makes our argument even stronger since in this case the entry barriers are so strong that entry is not profitable to any firm because any entering firm would have to exit when another firm enters 84 Entry Deterrence We now turn to the strategic approach for explaining entry barriers We assume that initially there is one firm called the incumbent or the established firm that is a monopoly in a certain market In the second stage we assume that another firm called the potential entrant is entering the market if entry results in above normal profit Modifying Bains classifications of entry deterrence we use the following terminology Blockaded entry The incumbent is not threatened by entry no firm would find it profitable to enter even if the incumbent produces the monopoly output level Page 187 Deterred entry The incumbent modifies its behavior say by lowering price or expanding capacity in order to deter entry if prices are lowered then we say that the incumbent exercises limit pricing Accommodated entry Entry occurs and the incumbent firm modifies its action to take into account of entry that occurs Thus blockaded entry corresponds to what we called entry barrier in section 83 where we discussed several conditions yielding entry barriers other than the behavior of incumbent firms In contrast we refer to entry deterrence and entry accommodation as actions taken by incumbent firms when faced with a threat of entry Earlier authors held that an incumbent firm may be able to deter entry by overproducing and selling at lower prices prior to the date at which entry is expected These types of models relied on the BainSylos postulate under which the prospective entrant was assumed to believe that the established firm would maintain the same output after entry that it did before entry Then the established firm naturally acquired a leadership role as described in the LeaderFollower model section 62 In addition some of the earlier models assumed that entrants have to sink output independent costs in order to begin their operation whereas incumbents do not Presently most economists disregard these arguments for the following reasons First note that this cost asymmetry could be reversed considering the fact that established firms may have to pay some costs that the entrant does not have to bear For example established firms may operate according to longterm contracts Most notably wage contractees and unions are hard to negotiate with and the downward adjustment of wages needed to meet the competition with the entrant would invoke tough resistance from workers and unions Yet in some instances the potential entrant is free to choose workers and can decide on wages without having any prior obligation The same argument holds for subcontracting and binding contracts with suppliers of raw material and parts In addition assuming asymmetric cost structure turns the problem of entry deterrence into an ad hoc problem since there always exists a level of entry cost that would prevent firms from entering the market Moreover even if the above asymmetry holds true in reality it is likely that in the long run the entrant would be able to collect a high enough duopoly profit to more than cover the entry cost In addition banks observing that the entering firm would make such a profit would be willing to lend the entrant the entry cost since the firm would be able to pay back the loan and interest with its future profits Second Friedman 1979 and Dixit 1980 question the validity of the BainSylos postulate by raising some doubt regarding the logic be Page 188 hind the above entrydeterrence argument They point out that the preentry price choice or quantity in our case of the established firm is irrelevant for the entry decision of the potential entrant The only thing that should matter to the potential entrant is what the postentry market structure would be After entry occurs and the entry cost is already paid there is no reason to assume that the firms would play the LeaderFollower game It would be more reasonable to assume that the firms would play Cournot or Bertrand where the firms have equal power and knowledge Now given that the entrant knows that the market structure would change after entry occurs all the firstperiod entry deterrence strategies limit pricing or overproduction are irrelevant to the postentry profits collected by all firms Third in modeling entry deterrence it is not clear why one firm gets to be the first to choose and commit itself to a certain production level thereby obtaining what is commonly called a firstmover advantage The approach to modeling entry deterrence based on the BainSylos postulate is given in subsection 841 where we sketch an analog to Spence 1977 and demonstrate that entry can be deterred if an incumbent firm builds an irreversible capacity prior to the period when entry is allowed so that a potential entrant faces a saturated market if it decides to enter Subsection 842 relaxes the Bain Sylos postulate and assumes that the incumbent is aware of the possibility that the entrant may find it profitable to alter its actions after entry occurs Subsection 843 Investment in capital replacement introduces a dynamic entrydeterrence model showing how in the face of entry threats an incumbent with depreciating capital is forced to invest more frequently than what is needed to simply replace depreciated capital Subsection 844 Judo economics focuses on the strategic choices of a potential entrant when an incumbent firm may find it more profitable to allow a smallscale entry rather than fighting it Subsection 845 Credible spatial preemption analyzes an incumbent differentiatedgood producer facing entry in one of its markets We conclude our analysis of entry barriers with subsection 846 where we demonstrate that limit pricing can serve as an entrydeterring strategy when the entrant does not know the production cost of the incumbent 841 Capacity commitment under the BainSylos postulate Earlier models analyzing entry deterrence adopted the BainSylos postulate under which the prospective entrant was assumed to believe that the incumbent firm would maintain the same output after entry as before Spence 1977 explicitly distinguishes between capacity and quantity produced In his model the quantity produced is constrained by Page 189 the amount of capacity firm 1 invests in the first period Thus as long as entry does not occur the capacity is underutilized However in the event of a threat of entry the incumbent can expand its output level and use all the capacity thereby reducing the price to the level that makes entry unprofitable In this subsection we refrain frommaking the distinction between capacity and output level and concentrate on analyzing how the incumbent determines how much capital to invest under the threat of entry Consider the twoperiod LeaderFollower game described in section 62 However instead of assuming that firms decide how much to produce let us assume that the firms actions are confined to how much capacity or capital to accumulate invest Although this distinction is only a semantic one it makes our story somewhat more convincing since capacity bears the sense of irreversibility one is unable to discard it and to collect the costs already paid thereby making capacity accumulation a credible strategic variable Thus in period 1 firm 1 has to choose its capacityoutput investment in period 2 firm 2 chooses whether to enter choosing k2 0 or to stay out k2 0 We assume that the firms are identical in all respects except that the potential entrant firm 2 has to pay an entry cost Such costs include an investment in new equipment payments to lobbyists for facilitating the industrys control regulations and so on We denote the entry cost by E To completely describe the game we define the profit of the firms collected at the end of the second period to be We solve this game backwards by first analyzing the last period given the action taken in the preceding period The second period In the second period firm 2 takes as given and chooses k2 to maximize its profit given in 816 There can be two cases Firm 2 enters and pays the entry cost E or it does not enter Suppose for a moment that it enters Then firm 2 chooses k2 to satisfy Page 190 Substituting into the profit function of firm 2 816 we have it that if firm 2 enters then which is greater than zero if and only if We summarize the analysis for the second period by the bestresponse function of firm 2 The first period In the first period fir I has to set k1 knowing how it will affect thecapacity choice of firm 2 That is firm 1 calculates 818 Firm I also knows that the bestresponse function of firm 2 is discontinuous when it sets Thus firm I would take into consideration that small changes in its capacity around may induce firm 2 to alter its entry decision With this discontinuity in mind our search for the profitmaximizing strategy for firm 1 would involve comparing the profit of firm 1 when firm 2 enters the leaders profit level denoted by with the profit of firm 1 when firm 2 does not enter the monopoly profit level denoted by Formally these profit levels are given by Thus for a given k1 the monopolys profit level is twice the leaders profit levels in the present formulation The two profit functions are drawn in Figure 84 In Figure 84 the upper bellshaped curves are the incumbents monopoly profit when entry does not occur The lower bellshaped curves are the leaders profit level when entry occurs Also the entrydeterring capacity level of firm 1 given by is marked by the vertical solid line with a rightward pointing arrow indicating that for firm 1 is a monopoly hence the upper bellshaped profit curves apply Figure 84 is divided into three parts indicating how firm 1 reacts for different levels of firm 2s entry cost 1 Blockaded entry This case is not displayed in Figure 84 but applies when high entry cost In this case choosing the monopoly capacity level is sufficient for deterring entry That Page 191 Figure 84 Incumbents profit levels and capacity choices for different levels of entry cost is when the entry cost is high firm 2 will not enter when firm 1 plays its monopoly capacity level Thus substituting k2 0 into 816 firm 1 chooses k1 to maximize its monopoly profit The first order condition satisfies Hence k1 12 Now for having this output level deterring the entry of firm 2 818 implies that E has to satisfy implying that 2 Indifference between deterrence and accommodation We need to find the magnitude of the entry cost parameter E that would make firm 1 indifferent to whether it chooses to set to deter entry or to set k1 12 and accommodate entry it is clear that k1 12 is the profitmaximizing capacity under monopoly as well as when entry occurs since both profit curves peak at k1 12 Thus we need to compare the leaders profit level under k1 12 when entry occurs given in 819 to the profit level when firm 1 deters entry by setting denoted by Hence Page 192 Thus we need to solve yielding implying that 3 Entry deterrence From case 2 and case 3 of Figure 84 we have it that entry deterrence is profitable for firm 1 when the entry cost is at an intermediate level That is when 000536 E 00625 4 Entry accommodation When the entry cost is very low firm 1 would have to increase k1 to a very high level in order to deter entry Case 4 of Figure 84 shows that if E 000536 deterring entry is not profitable and that entry accommodation yields a higher profit level for firm 1 842 Relaxing the BainSylos postulate So far our analysis has relied on the BainSylos postulate under which the potential entrant is assumed to believe that the incumbent firm will maintain the same action after entry as before Thus under this postulate the potential entrant is assumed to believe that upon entry the incumbent will utilize its entire capacity to produce the highest possible output level in order to make entry unprofitable for the entrant In this section following Dixit 1980 we demonstrate that such an assumption is inconsistent with a strategic behavior under a subgame perfect equilibrium Definition 210 on page 27 More precisely we demonstrate that under a subgame perfect equilibrium the incumbent firm will not find it profitable to utilize its entire capacity even when entry does occur Thus a rational potential entrant should be able to predict that a profitmaximizing incumbent will not find it profitable to utilize all its entire capacity Therefore we show that in a subgame perfect equilibrium a profitmaximizing incumbent will not invest in excess capacity for the purpose of entry deterrence In other words overaccumulation of capacity will not occur Consider the following twostage game In the first stage firm 1 incumbent chooses a capacity level that would enable firm 1 to produce without cost units of output in the second stage of the game If however the incumbent chooses to expand capacity beyond in the second stage then the incumbent incurs a unit cost of c per each unit of output exceeding Figure 85 illustrates the marginalcost function facing the incumbent in the second stage of the game Intuitively speaking we can say that any amount produced above the firms capacity will require special inputs that are costly to the firm Page 193 Figure 85 Capacity accumulation and marginal cost when purchased at the last minute Finally to make our argument even stronger we assume that capacity accumulation in the first stage is costless to the incumbent The entrant is assumed to make its entry decision in the second stage of the game More precisely in the second stage both firms jointly choose their output levels and play a Cournot game see section 61 We assume that firm 2 does not have any capacity and thus bears a unit cost of c which is the same unit cost of the incumbent for producing beyond its capacity If firm 2 chooses q2 0 we say that entry does not occur The game is illustrated in Figure 86 Figure 86 Relaxing the BainSylos postulate We now turn to the second stage after firm 1 has chosen its irrevocable capacity level given by Figure 87 illustrates Cournot output bestresponse functions for three given choices of by firm I in the first stage The bestresponse functions drawn in Figure 87 are derived in the same way as that under the conventional Cournot market structure Page 194 Figure 87 Bestresponse functions with fixed capacity Left low capacity Middle medium capacity Right High capacity see section 61 in particular Figure 61 on page 100 The only difference between the present case and the conventional Cournot case is that the incumbents bestresponse function is discontinuous at an output level reflecting a jump in the unit cost associated with a production level beyond the firms planned capacity Figure 87 has three drawings associated with having the incumbent investing in low medium and high capacity in the first period thereby determining three Cournot equilibria denoted by E1 E2 and E3 respectively The most important observation coming from Figure 87 is that the equilibrium marked by E2 is identical to the equilibrium marked by E3 despite the fact that Es is associated with a higher capacity level invested in by firm 1 in the first stage This proves our main proposition Proposition 87 The incumbent cannot deter entry by investing in a large capacity More generally investing in excess capacity cannot serve as a tool for deterring entry More interestingly in our example the firstperiod cost of capital capacity is zero Despite that cost firm 1 cannot benefit by investing in units of capital since after entry occurs the incumbents best response is to produce That is the entrant can calculate that in the subgame of the second period in a Cournot equilibrium firm 1 will limit its production for the same reason that any firm limits its production under a Cournot market structure preventing a price fall and will therefore enter The main message conveyed by Proposition 87 is that investing in excess capital cannot provide the incumbent with a credible threat by which convince the potential entrant that entry is unprofitable Thus the BainSylos postulate imposes an unrealistic belief on the potential Page 195 entrant namely the belief that the incumbent will utilize all its capacity after entry occurs despite the fact that this action does not maximize the incumbents profit 843 Investment in capital replacement So far we have assumed that investment in capacity is sufficient to produce output for the desired period of production However plants and equipment are of finite duration If investment in capital deters entry then entry is unavoidable if capital depreciates and the incumbent does not invest in capital replacement In what follows we construct a discretetime version of the analysis found in Eaton and Lipsey 1980 and investigate how the threat of entry affects the frequency of capital investment by an incumbent firm in the presence of depreciating capital Consider an industry with two firms firm 1 incumbent and firm 2 potential entrant Each firm can produce only if it has capital The profit of each firm is as follows If only firm 1 has capital in a certain period then firm 1 earns a monopoly profit given by H in this particular period If both firms have capital in a certain period then each earns a duopoly profit given by L in this period Suppose that in each period t t 0 1 2 each firm can invest F in capital with finite duration and that during the time periods of this capital the firm can produce any amount of a homogeneous product We denote the action taken by firm i in period t by where Invest or Not Invest Figure 88 illustrates the time path and the timing of actions taken by the two firms Figure 88 Capital replacement and entry deterrence In Figure 88 firm 1 the incumbent is assumed to invest in capital in period t 1 and then the game starts at t 0 where both firms can Page 196 invest in capital in any period t 0 1 2 3 We make the following assumption on the parameters of the model Assumption 81 1 Capital lasts for exactly two periods only At the end of the second period the capital completely disintegrates cannot be resold and has a scrap value of zero 2 The duopoly profit is insufficient to sustain two firms in the industry whereas the monopoly profit level is sufficiently high relative to the capital investment cost Formally 2L F H Assumption 81 implies that if firm I invests in capital in period t then using this capital the firm can produce in periods t and t 1 where the capital completely disintegrates at the end of the second period after production is undertaken The game proceeds as follows In period 0 if firm 2 potential entrant invests in capital then each firm earns L in period 0 If firm 2 does not enter does not invest in capital then firm 1 earns H in period 0 Let 0 ρ 1 denote the discount parameter and assume that each firm maximizes the sum of its discounted profit given by where if only firm i has capital in period t and if both firms have capital in period t and if firm i invests in capital in period t and if no investment is undertaken by firm i in period t Our purpose is to demonstrate the following Proposition 88 Under Assumption 81 1 If firm 2 is not allowed to enter then firm I invests in capital in odd periods only That is 2 If firm 2 is allowed to enter and if the time discount parameter is sufficiently small and satisfies Page 197 then the following strategies constitute a subgame perfect equilibrium Definition 210 for this game Hence in this equilibrium entry is deterred by having firm 1 incumbent investing in each period Proof We look at equilibrium strategies where firm 1 invests in every t and firm 2 does not invest First observe that since firm 1 invests at t and still has capacity at t 1 if firm 2 deviates and invests at t it will earn L F at t L F at t 1 and H F in each period thereafter Firm 2 will not deviate ie will not invest at t if Secondly if firm 1 deviates ie ceases investing at t 1 then it has no capacity at t and firm 2 will earn H F at t Hence firm 2 will enter Thirdly if firm 1 stops investing at t 1 it will earn a profit of H in period t 1 and zero thereafter Thus in order for having firm 1 engaging in continuous investment it must be that Therefore the strategies specified in 822 constitute a Nash equilibrium when condition 821 holds Proposition 88 conveys the very idea that in order to deter entry the incumbent must carry out a costly activity which is investing in extra capital capital that is not needed for production purposes This idea was suggested earlier by Schelling 1960 where he argued that in games involving such conflicts a threat that is costly to carry out can be made credible by entering into an advanced commitment That is we showed that despite the fact that capital lasts for two periods an incumbent monopoly must invest in each period in order to make entry unprofitable for potential entrants If the incumbent neglects to invest in even one period the entrant can credibly cause the exit of the incumbent by investing in capital Thus the fact that capital lasts for more than one period makes investing in capital a credible entrydeterring strategy because it ensures the existence of a firm in a subsequent period Page 198 844 Judo economics So far our discussion of entry deterrence has focused mainly on the incumbent firms In this subsection we analyze the strategic options available to the potential entrant prior to the time of entry into the industry In particular we analyze the entrants choice of capacity when facing a large dominant incumbent firm that has the option to expand capacity and deter entry We show that the potential entrant may profit by adopting a strategy of judo economics Gelman and Salop 1983 which refers to having the entrant invest in only limited capacitywhich would restrict the entrants scale of entry and therefore its market share We show that when the potential entrant limits its capacity sufficiently it is the incumbents best interest to accommodate entry rather than to fight it Consider a twostage game in which in the first stage a potentially entering firm chooses a whether to enter b its capacity maximum output level denoted by k and c its price denoted by pe In the second stage the incumbent firm chooses its price denoted by pI We assume that the incumbent firm is large in the sense that it has an unlimited capacity Assume that production is costless and that the firms produce a homogeneous product for a single market with a demand curve given by p 100 Q Also assume that all consumers prefer the less expensive brand however consumers prefer the incumbents brand at equal prices Formally let qI denote the quantity demanded from the incumbent firm and qe denote the quantity demanded from the entrant if entering Then for a given sufficiently low capacity invested by the entrant k the demand facing each firm is given by That is after the entrant sets pe the incumbent can always undercut the entrant by setting pI pe However if the incumbent sets a price slightly above the entrants price the entrant gets to sell the first k units and then the incumbent faces the residual demand given by qI 100 k pI Suppose now that in the first stage the entrant enters and sets a capacity k and a price pe Then in the second stage the incumbent can deter entry by setting pI pe or accommodate entry by setting pI pe If entry is deterred then the incumbents profit is given by In contrast if the incumbent accommodates entry then the incumbents profit is Thus under entry Page 199 accommodation the incumbent chooses pI pe to yielding a firstorder condition given by 0 100 k 2pI Therefore hence and Comparing the incumbents entrydeterring profit level to its profit under entry accommodation yields that Under entry accommodation the entrant earns πe pek 0 We now turn to the first stage where the entrant sets its capacity level and its price Figure 89 derived from 824 illustrates the range of k and pe that would induce the incumbent to accommodate entry Figure 89 Judo economics How an entrant secures entry accommodation Figure 89 demonstrates that for a sufficiently low pe there always exists k small enough to induce the incumbent to accommodate entry according to the condition given in 824 More precisely when the entrant reduces the price pe the horizontal line converges to the horizontal axis thereby increasing the area in which the incumbent accommodates the entry Thus Proposition 89 There exist a sufficiently limited capacity level k and a price pe set by the entrant that ensure that the incumbent will find it profitable to accommodate the entry The intuition behind this result is as follows When the incumbent accommodates the entrant it does not match the entrants price but Page 200 rather maintains an umbrella under which the entrant can prosper as long as it remains satisfied with its modest market share In this case the incumbent can maintain a higher price than the entrant and still sell since the entrant has a limited capacity that leaves a sufficiently profitable market share to the incumbent Thus when the entrant sets a sufficiently low capacity and price entry deterrence setting pI to a very low level yields a lower profit than entry accommodation to the incumbent firm The model presented in this subsection applies only to those situations in which the entrant can make credible capacitylimitation commitments Such credibility can be enhanced by the use of contracts For example entry accommodation is sometimes observed in the airline industry where large established airline firms accommodate small carriers on some routes after observing that the entrant purchased a limited number of airport gates a limited aircraft fleet and lowcapacity aircraft Of course as happens from time to time some of these small entrants grow to become major carriers 845 Credible spatial preemption Our entrydeterrence analysis has concentrated so far on entry in a single market for a homogeneous product In reality firms produce differentiated substitutable brands so entry is likely to cause a headtohead competition only on a subset of the incumbents already produced brands For example in the airline industry where a monopoly airline is threatened by entry it is likely to occur on a subset of the routes operated by the incumbent airline The question raised in Judd 1985 is how would the incumbent firm react to partial entry when entry into one market would affect the demand in a market for a substitute good hence the incumbents profit from the substitute good We demonstrate this entry problem by considering a monopoly firm firm 1 which owns two restaurants one Chinese denoted by C and one Japanese denoted by J Suppose that there are two consumers in town who are slightly differentiated with respect to the utility the receive from Chinese and Japanese food More precisely the utility of the consumer who is oriented toward Chinese food UC and the utility of the consumer who is oriented toward Japanese food UJ are given by where β reflects the satisfaction from eating and λ 0 denotes the slight Page 201 disutility a consumer has from buying his less preferred food We assume that λ β 2λ and normalize the restaurants costs of operation to zero Suppose first that both restaurants are owned by a single firm firm 1 Then 825 implies that the monopoly owner would charge prices pC pJ β in each restaurant and the monopolys total profit would be π1 2 β Entry into the market for Chinese food Suppose that a new restaurant firm 2 with a different owner opens a new Chinese restaurant that serves food identical to the already existing Chinese restaurant owned by the monopoly Assuming price competition we see the price of Chinese food drop to zero the assumed unitproduction cost Thus How would entry into the Chinese food market affect the price of a Japanese dinner Well clearly if the monopoly does reduce its price of a Chinese dinner to zero all consumers including the one oriented toward Japanese food would purchase only Chinese food Therefore the maximum price the monopoly Could charge for a Japanese dinner would be pJ λ Clearly for this price the consumer oriented toward Japanese food would purchase Japanese since That is at pJ λ the Japanesefoodoriented consumer is indifferent to whether he or she buys Japanese gaining a utility of UJJ or Chinese gaining UJC In this case the profit earned by the monopoly after the entry into the Chinesefood market occurs is π1 λ Incumbent withdraws from the Chinese restaurant Now suppose that firm 1 the initial monopoly on oriental food shuts down its Chinese restaurant and keeps only the Japanese restaurant In this event after entry occurs there are two restaurants one serving Chinese food and the other serving Japanese food Thus the market structure is now a duopoly with firms selling differentiated products Lemma 81 The unique duopoly price game between the Chinese and the Japanese restaurants results in the consumer oriented toward Japanese food buying from the Japanese restaurant the consumer oriented toward Chinese food buying from the Chinese restaurant and equilibrium prices given by Proof We have to show that no restaurant can increase its profit by undercutting the price of the competing restaurant If the Japanese Page 202 restaurant would like to attract the consumer oriented toward Chinese food it has to set pJ pC λ β λ In this case π2 2β λ However when it does not undercut π2 β 2β λ since we assumed that β 2λ A similar argument reveals why the Chinese restaurant would not undercut the Japanese restaurant We can now state our major proposition Proposition 810 When faced with entry into the Chinese restaurants market the incumbent monopoly firm would maximize its profit by completely withdrawing from the Chinese restaurants market Proof The profit of the incumbent when it operates the two restaurants after the entry occurs is π1 λ If the incumbent withdraws from the Chinese restaurant and operates only the Japanese restaurants Lemma 81 implies that π1 β λ The intuition behind Proposition 810 is as follows When entry occurs in one market the price falls to unit cost Given the reduction in this price consumers buying a substitute good Japanese food would switch to buying Chinese food Hence the incumbent would have to reduce the price in its other market despite the fact that no entry occurred in the other market Consequently the incumbent would suffer a profit reduction in both markets To avoid the latter the incumbent would benefit from withdrawing and letting the entrant charge a higher price in the competing market This would enable the incumbent to maintain the monopoly price in the remaining monopolized market Japanese food Thus by withdrawing from competition the incumbent differentiates itself from the entrant so both firms could maintain a high price 846 Limit pricing as cost signaling Friedmans argument concerning the irrelevance of limit pricing raises the question whether incumbent firms would ever find it useful to exercise limit pricing during the preentry period Milgrom and Roberts 1982 came up with an argument that limit pricing or expanded capacity or quantity produced can serve as a costsignaling device to the potential entrant who may not know the cost structure of the incumbent firm We discuss here a simplified version of their model Demand firms and timing There are two periods denoted by t 1 2 The market demand curve in each period is given by p 10Q where Q is the aggregate amount sold to consumers Firm 1 is the incumbent and has to choose an output level in period 1 denoted by Firm 2 does not exist in t 1 and chooses Page 203 whether or not to enter only in the second period Thus firm 1 earns profits in the preentry period t 1 and in t 2 What about the output levels in the second period Following Friedmans argument we assume the following Assumption 82 In the second period t 2 if entry occurs then both firms play the Cournot game If entry does not occur at t 2 firm 1 produces the monopoly output level This assumption highlights Friedmans argument in the sense that the incumbents action at t 1 has no influence on the market structure at t 2 and therefore we assume the most commonly used market structure for t 2 which is Cournot if entry occurs and monopoly in the case of no entry Cost and information Firm 2s unitproduction cost is given by c2 1 In addition firm 2 has to pay an entry cost of F2 9 if it enters at t 2 The cost structure of firm 2 is assumed to be common knowledge In contrast the cost structure of firm 1 the incumbent is known only to firm 1 The potential entrant does not exactly know the cost structure of the incumbent but it knows the probability distribution of cost functions Formally firm 2 knows that the unit cost of firm 1 satisfies That is firm 2 bases its decisions on the assumption that with 50 probability the incumbent is a lowcost firm c1 0 and with a 50 probability the incumbent is a highcost firm c1 4 Profits The incumbent collects profits in periods I and 2 and maximizes the sum of the two periods profits The entrant collects profit only in the second period In section 61 you have learned how to calculate the Cournot profit levels so we avoid performing these simple calculations These profit calculations are snmmarized in Table 82 The twoperiod game In the preentry era period 1 firm 1 chooses its output level Thus the profit of firm 1 in t 1 is In period 2 firm 2 observes and decides whether or not to enter Its decision is based on the value of and on the estimated cost structure of firm 1 given in 826 Figure 810 illustrates this game Page 204 Incumbents cost Firm 2 potential entrant ENTER DO NOT ENTER Low c1 0 Highc1 0 Table 82 Profit levels for t 2 depending on the entry decision of firm 2 Note All profits are functions of the cost of firm 1 c1 is the monopoly profit of firm 1 is the Cournot profit of firm i i 1 2 Figure 810 Twoperiod signaling entrydeterrence game Solving the game assuming a highcost incumbent Without any further knowledge and assuming that firm 2 maximizes expected profit we learn from 826 and Table 82 that upon entry firm 2s expected profit is hence with no additional knowledge firm 2 would enter But why shouldnt the incumbent always state that it is a lowcost firm rather than a highcost firm Well the incumbent can state whatever it wants but firm 2 has no reason to believe the incumbents statements Hence given that entry occurs and the firms play Cournot in t 2 the best firm 1 could do is to maximize the firstperiod profit by playing the monopolys output in t 1 That is to set Thus if the incumbent is a highcost firm it would not attempt to limit its price and entry will occur Page 205 Solving the game assuming a lowcost incumbent Suppose that the incumbent firm 1 is a lowcost firm c1 0 Then if firm 2 were to know that firm I is a lowcost one Table 82 shows that it would not enter since entry yields But since firm 2 does not know for sure that firm I is a lowcost one the incumbent has the incentive to reveal it to firm 2 The purpose of this model is to demonstrate how limit pricing or excess production can serve as a means by which firm I can signal to firm 2 that it is a lowcost firm thereby convincing firm 2 that entry is not profitable Proposition 811 A lowcost incumbent would produce and entry will not occur in t 2 Sketch of Proof In order for the incumbent to convince firm 2 that it is indeed a lowcost firm it has to do something heroic More precisely in order to convince the potential entrant beyond all doubts that firm 1 is a lowcost one it has to do something that a highcost incumbent would never donamely it has to produce a firstperiod output level that is not profitable for a highcost incumbent Now a highcost incumbent would not produce since That is a highcost incumbent is better off playing a monopoly in the first period and facing entry in the second period than playing in the first period and facing no entry in t 2 Finally although we showed that indeed transmits the signal that the incumbent is a low cost firm why is the incumbents profitmaximizing output level given that the monopolys output level is much lower Clearly the incumbent wont produce more than 583 since the profit is reduced gets higher above the monopoly output level Also 828 shows that any output level lower than 583 would induce entry and given that entry occurs the incumbent is best off playing monopoly in t 1 Hence we have to show that deterring entry by producing yields a higher profit than accommodating entry and producing the monopoly output level in t 1 That is hence a lowcost incumbent will not allow entry and will not produce Page 206 847 Other entrydeterrence methods The literature on entry deterrence explores various entrydeterring actions taken by incumbent firms see survey articles by Neven 1989 and Wilson 1992 One possible action referred to as raising a rivals cost is analyzed by Salop and Scheffman 1983 They suggest that incumbent firms may possess a variety of methods for raising the cost of entering firms For example one way of doing that is for the incumbent firm to sign high wage contracts thereby raising the industrys labor cost Another is for the incumbent to lobby for higher tax rates As noted earlier potential entrants may be immune from these entrydeterring strategies since they may not be subjected to binding wage and other cost contracts Note that in order for these actions to constitute entrydeterring methods one needs to show that the these methods do not result in having the incumbent going bankrupt Another possible action analyzed in Aghion and Bolton 1987 suggests that incumbent firms rush to sign contracts with buyers in order to preempt entry Gallini 1984 suggests that an incumbent can minimize its loss to firms producing potentially more advanced brands by simply licensing their own older technologies to potential entrants The idea is that without licensing potential entrants would develop superior technologies that would wipe out producers of older technologies Finally Spiegel 1993 demonstrates that incumbent firms can deter entry by subcontracting with other incumbent firms producing competing brands Intuitively if those firms have different cost structure horizontal subcontracting reduces average costs of the incumbent firms thereby reducing the likelihood that entry will occur Another way in which entry can be deterred is for the incumbent to deny access to a new technology by acquiring a patent right for its technology see Gilbert and Newbery 1982 Finally Scherer 1979 and Schmalensee 1978 analyze the FTC complaint that the four major cereal producers managed to deter entry by proliferating product varieties thereby leaving insufficient room for the entry of new brands Their result stems from the assumption of that the incumbents decision to produce a brand is irreversible however subsection 845 demonstrates that incumbents may be better off to withdraw from the production of some brands in the presence of entry rather than fighting it 85 Contestable Markets Baumol Panzar and Willig 1982 proposed a market structure that describes the behavior of incumbent firms constantly faced by threats of entry The main assumption underlying this market structure is that Page 207 entry does not require any sunk cost Note that with the absence of sunk cost incumbent firms are subject to a hitandrun entry meaning that potential entrants can costlessly enter and exit the industry without having to wait until they generate a sufficient amount of revenue to recover the sunk cost of entry Therefore if incumbent firms do not have any cost advantage over potential entrants a contestable market equilibrium will result in having an incumbent firm making only normal zero profit Assume that in a homogeneous product industry there is one incumbent firm facing entry by potential competitors Let all firms have identical and increasing returnstoscale technologies summarized by the cost function TCqi F cqi and assume that the inverse aggregate demand facing the industry is given by p a Qd Definition 81 1 An industry configuration is the incumbents pair pIqI of price charged and quantity produced 2 An industry configuration is said to be feasible if a At the incumbents price pI the quantity demanded equals the incumbents quantity supplied That is if pI a qI b The incumbent makes a nonnegative profit That is 3 An industry configuration is said to be sustainable if no potential entrant can make a profit by undercutting the incumbents price That is there does not exist a price pe satisfying and a corresponding entrants output level qe satisfying such that pe qe F cqe 4 A feasible industry configuration is said to be a contestablemarkets equilibrium if it is sustainable Thus an industry configuration is sustainable if no other firm could make a strictly positive profit by setting a lower or equal price while producing no more than the quantity demanded by the consumers A contestablemarket equilibrium is illustrated in Figure 811 where the price pI and quantity produced qI satisfy the consumers aggregate demand curve and in addition lie on the incumbents average totalcost function thereby ensuring that the incumbent does not incur a loss Hence this configuration is feasible Now given that all firms share the same cost structure it is clear that under the industry configuration illustrated in Figure 811 no other Page 208 Figure 811 Contestablemarkets equilibrium firm could lower the price and make a strictly positive profit Hence this configuration is sustainable It should be noted that the contestable market structure can be used to describe an industry comprised of multiproduct firms that is firms producing a variety of different products see Baumol Panzar and Willig 1982 Indeed the advantage of using the contestablemarkets market structure is that it can be applied to more realistic industries where firms produce more than one product especially since all other market structures are defined for singleproduct firms that are rarely observed The contestablemarket equilibrium defined in Definition 81 relies on the assumption that firms do not incur any sunk cost upon entry and therefore can costlessly enter and exit the industry This assumption is problematic since it is hard to imagine an industry where firms do not have to sink any irrevocable investment prior to entry That is firms generally conduct a market survey place advertisements and pay fees prior to entry and these costs are definitely sunk and cannot be recovered Moreover Stiglitz 1987 pointed out the significance of this assumption by showing that if entrants face even tiny sunk costs prior to entry then the only subgame perfect equilibrium is an incumbent charging a monopoly price and making a monopoly profit In other words although the contestablemarket equilibrium yields a result that the incumbent makes zero profit Proposition 85 showed that introducing even tiny sunk cost would imply that in a subgame perfect equilibrium of an entrydeterrence game the incumbent makes pure monopoly profit Therefore the sensitivity of the market outcome to the existence of even small sunk cost is highly problematic because these two models have contradictory policy recommendations On the one hand contestablemarket equilibrium implies that no intervention is needed Page 209 by the antitrust authorities since even a single firm would charge the socially efficient price On the other hand the introduction of even a small sunk costs turns our model into a sustained monopoly one which the antitrust authority would like to challenge Finally Schwartz 1986 has shown that despite the assumed easy exit of potential entrants hit and run entry is unprofitable if incumbents price responses are sufficiently rapid 86 Appendix Merger and Antitrust Law Section 7 of the Clayton Act 1914 states that No person engaged in commerce or in any activity affecting commerce shall acquire directly or indirectly the whole or any part of the stock or other share capitalshall acquire the whole or any part of the assets of another person engaged also in commerce or in any activity affecting commerce where in any line of commerce in any section of the country the effect of such acquisition may be substantially to lessen competition or to tend to create a monopoly Section 7 of the Clayton Act amended in 1950 was needed because Sections I and 2 of the Sherman Act 1890 were not sufficient to halt mergers that would increase concentration and would reduce competition Of course a question remains about why an increase in concentration would reduce competition and raise prices This idea is built on two premises First that collusion or tacit coordination is less likely to succeed in less concentrated markets where price cuts are less likely to be noticed by rival firms and second that antitrust should be viewed as consumer protection and that consumers tend to lose when faced by monopoly sellers The discussion in this section is divided into two parts We first discuss the procedure by which the FTC Federal Trade Commission and the DOJ Department of Justice can intervene in order to challenge a merger Then we proceed to the details that the two agencies use to measure the effect of a merger The interested reader is referred to Asch 1983 Fisher 1987 Gellhorn 1986 Salop 1987 and White 1987 for further reading and more references 861 Challenging a merger The monitoring of merger activities is in the hands of the FTC and the DOJ The FTC issues guidelines to the DOJ recommending what types of mergers should be challenged It is important to note that these guidelines do not constitute a law but rather recommendations Page 210 to the DOJ for starting to take actions against undesired mergers In practice firms with assets or sales in excess of 100 million must report acquisitions of assets valued in excess of 15 million A merger does not take place until the FTC or the DOJ determines the competition effects of such an acquisition With this procedure very few cases are brought to courts since in most cases the FTC evaluation is sufficient for providing the signals to the acquiring firm about whether it should proceed with the acquisition or call it off 862 Merger guidelines The purpose of horizontal merger guidelines is to describe the analytical process that the agencies will employ to decide whether to challenge a merger the guidelines are issued by the FTC and are suggestive rather than definitive Salop 1987 summarizes five criteria that characterize those used by the FTC and the DOJ for evaluating a proposed merger 1 the scope of the market upon which the merger may have anticompetitive effects 2 the effect on concentration 3 the ease of entry into the market 4 other factors related to the ease of collusion in the market and 5 efficiency gains such as cost reduction associated with the merger In 1982 the Reagan administration came up with new merger guidelines released in 1984 and modified in 1992 The scope of the relevant market was defined in price terms That is the relevant antitrust market is defined as a set of products and a geographical area where firms could profitably raise prices by at least 5 above the premerger price for at least one year These guidelines suggest that a merger should not be challenged if the postmerger HerfindahlHirshman concentration index IHH defined by 82 satisfies Thus a merger is more likely to be challenged when it results in a higher concentration ratio and when it results in a more significant change in concentration More precisely at low postmerger concentration levels a merger resulting in a change in the IHH of a less than 100 would not be challenged However at a high postmerger IHH a merger leading to a change of less than 100 but greater than 50 is likely to be challenged In the above ΔIHH measures the difference in the IHH measure before and after the proposed merger For example if firm 1 maintaining a market share s1 and firm 2 maintaining a market share of s2 merge Page 211 then the market share of the newly merged firm is expected to be s1 s2 In this case The higher the concentration is the more likely merger is to be challenged even if the merger causes only a small increase in the degree of concentration Several authors for example Farrell and Shapiro 1990 and those found in their references have criticized the use of the IHH as a reliable measure of a mergerinduced change in concentration because it assumes that the merged firms maintain the exact sum of the market shares the merged firms had prior to the merger However it is likely that the sum of the market shares of the merged firm would fall after the merger in the case where entry barriers do not prevail Finally in 1992 the DOJ and the FTC released modified horizontal merger guidelines see Department of Justice and Federal Trade Commission Horizontal Merger Guidelines April 2 1992 The release marks the first time that the two federal agencies that share antitrust enforcement jurisdiction have issued joint guidelines The new guidelines reflect the experience of the DOJ and the FTC in applying the 1984 merger guidelines The 1992 guidelines modify the test for identifying the relevant market The 1984 guidelines hypothesized a uniform price increase to identify the market Under the 1992 guidelines the price increase is not necessarily uniform Instead the new guidelines assume that a hypothetical monopolist may increase prices for some localities more than for others Similar to the 1984 guidelines a post merger concentration level of IHH 1000 classifies the market in the region as unconcentrated A post merger concentration of is regarded as moderately concentrated Mergers producing ΔIHH 100 raise significant competitive concerns depending on the factors set forth in Sections 25 of the 1992 guidelines Post merger concentration level IHH 1800 is regarded as highly concentrated Mergers yielding a change in concentration raise significant competitive concerns depending on the factors set forth in Sections 25 of the 1992 guidelines Mergers yielding ΔIHH 100 are regarded as likely to create or enhance market power or facilitate its exercise This presumption may be overcome by showing that the factors set forth in Sections 25 of the 1992 guidelines make it unlikely that the merger will enhance market power Sections 25 consider potential adverse competitive effects of mergers in addition to market concentration measured by the IHH These effects include i the likelihood of coordination among firms ii conditions revealing implicit or explicit coordination such as common price Page 212 fixed price differentials stable market shares or consumer or territorial restrictions iii detection of conditions making punishments on deviations from collusion more effective thereby increasing the likelihood of collusion iv the likelihood that a merger between firms distinguished by differentiated products to cause a price increase for all differentiated brands v ability of rival sellers to replace lost competition 87 Appendix Entry Deterrence and Antitrust Law Singlefirm conduct is covered by Section 2 of the Sherman Act 1890 under which it would be a violation of the antitrust law for an incumbent firm to engage in actions that would limit competition as stated in Section 2 of the Sherman Act 1890 Every person who shall monopolize or attempt to monopolize or combine or conspire with any other person or persons to monopolize any part of the trade or commerce among the several States or with foreign nations shall be deemed guilty of a felony Thus Section 2 focuses on the unilateral conduct of a firm whereas Section 1 focuses on the duality of actions among firms More precisely the essence of an offense under Section 1 is the act of joining together to conspire to limit competition and therefore the main concern is to find an agreement among firms In contrast Section 2 is concerned with the act of a monopoly that misuses its power by taking exclusionary actions Predatory prices are condemned but there is little agreement on what defines predatory prices A proof of pricing below average cost constitutes predatory pricing and in this case the burden of proof is on the defendant to show that either the prices are not below average cost or that the low prices are temporary for promotional reasons only However prices that exceed average cost can still be considered as predatory if they are set in order to eliminate competition with other firms since any attempt to monopolize is a felony under Section 2 Another violation of Section 2 is a refusal to deal which refers to cases where a monopoly at one level of a chain of distribution refuses to deal with the next level in order to gain a monopoly position at both levels Finally product innovation is not considered to be a violation of Section 2 even if the introduction of the new product into the market makes it difficult for other firms to compete or even survive We conclude by discussing how the FTC handles anticompetitive behavior of incumbent firms Section 5a1 of the Federal Trade Cornminion Act 1914 states Page 213 Unfair methods of competition in or affecting commerce and unfair or deceptive acts or practices in or affecting commerce are declared unlawful In earlier periods after the FTC was established the FTC concentrated on promoting fair trade practices among trade associations Over the years the FTC extended its role in enforcing these laws by conducting repeated investigations for the purpose of finding violations of firms that use a variety of anticompetitive methods described earlier in the chapter in order to maintain their dominance in the market When the FTC suspects a violation it opens an investigation against the suspected firms and looks at the products design and its distribution channels in order to find a clue about whether these activities deter potential entrants from entering into the market Investigations of these kinds are generally made public and by themselves encourage more firms to enter the market with competing brands knowing that predatory activities will not be sustained 88 Exercises 1 The bicycle industry consists of seven firms Firms 1 2 3 4 each has 10 market share and firms 5 6 7 each has 20 market share Using the concentration measures defined in Section 81 answer the following questions a Calculate I4 for this industry b Calculate the IHH for this industry c Now suppose that firms 1 and 2 merge so that the new firm will have a market share of 20 i Calculate the post merger IHH ii Calculate the change in the IHH caused by the merger That is calculate ΔIHH iii Using the merger guidelines described in subsection 862 evaluate the proposed merger and predict whether this merger will be challenged or not Explain 2 In an industry there are three firms producing a homogeneous product Let qi denote the output level of firm i i 1 2 3 and let Q denote the aggregate industryproduction level That is Q q1 q2 q3 Assume that the demand curve facing the industry is p 100 Q Solve the following problems a Find the Cournot equilibrium output and profit level of each firm b Now suppose that firms 2 and 3 merge into a single firm that we call firm 4 Calculate the profit level of firm 4 under a Cournot market structure Page 214 c Do firms 2 and 3 benefit from this merger d Now suppose that firm 1 merges with firm 4 Does firm 4 benefit from the merger with firm 1 e Explain why the first and the second mergers yield different results regarding the profitability of mergers 3 Consider the merger among firms producing complementary components studied in subsection 823 Suppose that consumers desire computer systems composed of one computer denoted as product X and two diskettes denoted as product Y Thus our consumers treat computers and diskettes as perfect complements where for each computer the consumers need two diskettes Let px denote the price of a computer and pY denote the price of a single diskette Thus the price of a computer system is ps pX 2pY Formally let the demand function for computer systems be given by Answer the following questions assuming that production is costless a Suppose that the X producer and the Y producer are independent Solve for the Nash Bertrand equilibrium in prices Calculate the equilibrium prices the quantity produced of each product and firms profit levels b Now suppose that firms X and Y merge under a single ownership Calculate the monopoly equilibrium prices the quantity produced of each product and the monopolys profit c Is this merger welfareimproving Compare system prices and profits of the firms before and after the merger 4 Consider the contestablemarkets market structure defined in section 85 Suppose that in the industry there is one incumbent firm and several potential competitors all having identical technologies summarized by the cost function TCqi 100 qi2 where qi is the output of firm i Solve for a contestablemarkets equilibrium assuming that the inverse aggregate demand facing the industry is given by p 60 4Qd 89 References Aghion P and P Bolton 1987 Contracts as a Barrier to Entry American Economic Review 77 388401 Asch P 1983 Industrial Organization and Antitrust Policy New York John Wiley Sons Bain J 1956 Barriers to New Competition Cambridge Mass Harvard University Press Bain J 1972 Essays on Price Theory and Industrial Organization Boston Little Brown Page 215 Baumol W J Panzar and R Willig 1982 Contestable Markets and the Theory of Industry Structure New York Harcourt Brace Jovanovich Davidson C and R Deneckere 1984 Horizontal Mergers and Collusive Behavior International Journal of Industrial Organization 2117132 Dixit A 1979 A Model of Duopoly Suggesting a Theory of Entry Barriers Bell Journal of Economics 10 2032 Dixit A 1980 The Role of Investment in EntryDeterrence Economic Journal 90 95106 Eaton B C and R Lipsey 1980 Exit Barriers are Entry Barriers The Durability of Capital as a Barrier to Entry Bell Journal of Economics 11 721729 Economides N and S Salop 1992 Competition and Integration Among Complements and Network Market Structure Journal of Industrial Economics 40 105123 Farrell J and C Shapiro 1990 Horizontal Mergers An Equilibrium Analysis American Economic Review 80 107126 Fisher F 1987 Horizontal Mergers Triage and Treatment Journal of Economic Perspectives 1 2340 Friedman J 1979 On Entry Preventing Behavior and Limit Price Models of Entry In Applied Game Theory edited by S Brains A Schotter and G Schwödiauer Würzburg PhysicaVerlag Gallini N 1984 Deterrence through Market Sharing A Strategic Incentive for Licensing American Economic Review 74 931941 Gaudet G and S Salant 1992 Mergers of Producers of Perfect Complements Competing in Price Economics Letters 39 359364 Gellhorn E 1986 Antitrust Law and Economics St Paul Minn West Publishing Co Gelman J and S Salop 1983 Judo Economics Capacity Limitation and Coupon Competition Bell Journal of Economics 14 315325 Geroski P R Gilbert and A Jacquemin 1990 Barriers to Entry and Strategic Competition Chur Switzerland Hardwood Academic Publishers Gilbert R and D Newbery 1982 Preemptive Parenting and the Persistence of Monopoly American Economic Review 72 514526 Jacquemin A 1987 The New Industrial Organization Cambridge Mass MIT Press Judd K 1985 Credible Spatial Preemption Rand Journal of Economics 16 153166 Milgrom P and J Roberts 1982 Limit Pricing and Entry Under Incomplete Information An Equilibrium Analysis Econometrica 50 443459 Page 216 Neven D 1989 Strategic Entry Deterrence Recent Developments in the Economics of Industry Journal of Economic Surveys 3 213233 Ordover J G Saloner and S Salop 1990 Equilibrium Vertical Foreclosure American Economic Review 80 127141 Perry M 1989 Vertical Integration Determinants and Effects In Handbook of Industrial Organization edited by R Schmalensee and R Willing Amsterdam NorthHolland Ravenscraft D and F M Scherer 1987 Mergers Selloffs and Economic Efficiency Washington DC The Brookings Institution Salant S S Switzer and R Reynolds 1983 Losses from Horizontal Merger The Effects of an ExogenousºChange in Industry Structure on CournotNash Equilibrium Quarterly Journal of Economics 98 185199 Salinger M 1988 Vertical Mergers and Market Foreclosure Quarterly Journal of Economics 77 345356 Salop S 1987 Symposium on Mergers and Antitrust Journal of Economic Perspectives 1 312 Salop S and D Scheffman 1983 Raising Rivals Cost American Economic Review 73 267 271 Schelling T 1960 The Strategy of Conflict Cambridge Mass Harvard University Press Scherer F M 1979 The Welfare Economics of Product Variety An Application to the ReadyTo Eat Cereals Industry Journal of Industrial Economics 28 113133 Schmalensee R 1978 Entry Deterrence in the ReadyToEat Breakfast Cereal Industry Bell Journal of Economics 9 305327 Schwartz M 1986 The Nature and Scope of Contestability Theory Oxford Economic Papers 38 Suppl 3757 Sonnenschein H 1968 The Dual of Duopoly in Complementary Monopoly Or Two of Cournots Theories are One Journal Political Economy 76 316318 Spence M 1977 Entry Capacity Investment and Oligopolistic Pricing Bell Journal of Economics 8 534544 Spiegel Y 1993 Horizontal Subcontracting Rand Journal of Economics 24 570590 Stiglitz J 1987 Technological Change Sunk Costs and Competition Brookings Papers on Economic Activity 3 Special Issue on Microeconomics 883937 Tirole J 1988 The Theory of Industrial Organization Cambridge Mass MIT Press von Weizsäcker 1980 A Welfare Analysis of Barriers to Entry RAND Journal of Economics 11 399420 Page 217 White L 1987 Antitrust and Merger Policy A Review and Critique Journal of Economic Perspectives 1 1322 Wilson R 1992 Strategic Models of Entry Deterrence In Handbook of Game Theory edited by R Aumann and S Hart Amsterdam NorthHolland Page 219 PART III TECHNOLOGY AND MARKET STRUCTURE Page 221 Chapter 9 Research and Development Innovation is an activity in which dry holes and blind alleys are the rule not the exception Jorde and Teece Innovation and Cooperation Implications for Competition and Antitrust Innovation is the search for and the discovery development improvement adoption and commercialization of new processes new products and new organizational structures and procedures Firms spend substantial amounts on research and development RD In the developed countries industries can be characterized according to the ratio of their RD expenditure to output sales Industries that exhibit high ratios include aerospace 23 office machines and computers 18 electronics 10 and drugs 9 Industries with RD expenditure to output ratios of less than 1 include food oil refining printing furniture and textiles OECD 1980 data So far in our analysis we have assumed that a production process or knowhow can be characterized by a welldefined production function or by its dual the cost function see section 31 Moreover we have assumed that the production function is exogenous to the firms and viewed as a black box by the producers In this chapter we analyze how firms can influence what is going on inside these black boxes by investing resources in innovation activities We then analyze the methods by which society protects the right of innovators in order to enhance innovation activities in the economy Research and development is generally classified into two types a process innovation the investment in labs searching for costreducing Page 222 technologies for producing a certain product and b product innovation the search for technologies for producing new products It is often argued that from a logical point of view there is no difference between the two types of innovation since product innovation can be viewed as a costreducing innovation where the production cost is reduced from infinity when the product was not available to a finite level However many intuitively believe that there is a difference The concept of RD is very difficult to understand and therefore to model since the act of doing RD means the production of knowledge or knowhow see Mokyr 1990 and Rosenberg 1994 for a historical overview of innovation and Dosi 1988 and Freeman 1982 a survey of the literature and empirical evidence of innovation Although we have so far always succeeded in avoiding discussion of the foundation of production functions and what knowhow is in this chapter we discuss precisely that by defining RD as the act of creating or changing the production functions Section 91 Process Innovation classifies two types of process innovation Section 92 Innovation Race analyzes how firms compete for discovering new technologies and evaluates whether the equilibrium RD level is below or above the socially optimal RD level Section 93 Cooperation in RD analyzes how RD is affected when firms coordinate their RD efforts Section 94 Patents analyzes how society encourages RD by granting patent rights to innovators and suggests a method for calculating the optimal duration of patents Section 95 Licensing an Innovation explains why firms tend to license their patented technologies to competing firms Section 96 International RD Races analyzes why governments subsidize RD for exporting firms In the appendix section 97 analyzes patent law from historical and legal perspectives Section 98 discusses the legal approach to cooperative RD 91 Classifications of Process Innovation This section classifies process costreducing innovation according to the magnitude of the cost reduction generated by the RD process Consider an industry producing a homogeneous product and suppose that the firms compete in prices ie Bertrand competition described in section 63 on page 107 Assume that initially all firms possess identical technologies meaning that they all produce the product with a unit production cost c0 0 Then initially there is a unique Bertrand equilibrium where all firms sell at unit cost p0 c0 make zero profits and produce a total of Q0 units of output This equilibrium is illustrated in Figure 91 Suppose now that one and only one firm has the following RD tech Page 223 Figure 91 Classification of process innovation nology the firm can construct a research lab engaging in costreducing innovation that leads to a unitcost technology of c c0 Now recalling from chapter 5 that the pure monopolys profit maximizing output and price can be found by equating MRQ c we distinguish between a large and a small costreducing innovation in the following way Definition 91 Let pmc denote the price that would be charged by a monopoly firm whose unit production cost is given by c Then 1 Innovation is said to be large or drastic or major if pmc c0 That is if innovation reduces the cost to a level where the associated pure monopoly price is lower than the unit production costs of the competing firms 2 Innovation is said to be small or nondrastic or minor if pm c c0 Figure 91 illustrates the two types of process innovation A cost reduction from c0 to c1 is what we call a small innovation That is the cost reduction is not large enough implying that the innovating firm does not charge the pure monopoly price In this case the innovating firm will undercut all its rivals by charging a price of and will sell Q0 units of output In other words a small innovation does not change the market price and the amount purchased by consumers The only consequence of a small innovation is that the innovator sells to the entire market and makes strictly positive profit equal to c0 c1Q0 Page 224 In contrast a cost reduction from c1 to c2 in Figure 91 illustrates a large costreducing innovation since the firm can undercut its rivals by simply charging the pure monopoly price associated with its new cost structure That is Thus a large innovation reduces the market price and increases quantity to Q2 Finally note that Definition 91 connects the physical change of cost reduction with the market conditions demand That is what we mean by small or large innovation depends on demand conditions and the market structure in addition to the cost reduction itself 92 Innovation Race The timing of innovation plays a crucial role in the marketplace There are two reasons why in most cases a firm that is first to discover a new technology or a new product gains an advantage over competing firms First the firm is eligible to obtain a patent protection that would result in earning monopoly profits for several years Second consumers associate the innovator with a higherquality producer and will therefore be willing to pay a higher amount for the brand associated with the innovator Given the significance of becoming the first to discover firms invest large sums RD knowing that not discovering or discovering too late may result in a net loss from the innovation process In this section we analyze the behavior of firms competing to discover a new product or a process and we focus on the following questions Do firms invest in RD more or less than the socially optimal level What is the impact of RD competition on the expected date when the new product will be produced and marketed to consumers Assume that the discovery translates into a prize that can be viewed as the value of a patent associated with several years of earning monopoly profits Consider a twofirm industry searching for a new technology for producing a new product The discovery of the product is uncertain Each firm k k 1 2 can engage itself in RD by investing an amount of I in a research lab The payoff from RD to a firm is as follows Assumption 91 Once a firm invests I in a lab it has a probability α of discovering a technology that yields a profit of V if the firm is the sole discoverer V2 if both firms discover and 0 if it does not discover 921 Equilibrium RD in a race We denote by Eπkn the expected profit of firm k from investing in innovation when the total number of firms engaging in similar RD is n Page 225 n 1 2 Also we denote by ik the investment expenditure of firm k A single firm undertaking RD If only firm 1 invests in RD the firm discovers with probability α therefore earning a profit of V I and does not discover with probability 1 α earning a negative profit given by I Therefore its expected profit is given by Eπ11 αV I Hence equating the expected profit to zero yields that the RD investment decision of firm 1 is given by Two firms undertake RD The twofirm technology race highlights two important uncertainties facing firms engaging in RD First there is technological uncertaintywhether or not the firm will discover the new product Second there is market uncertaintywhether or not the new product will be discovered by the rival firm When the two firms engage in RD the expected profit of each firm k is given by Equating 92 to zero implies that the following is a sufficient condition for having both firms profitably undertaking RD Figure 92 illustrates the two conditions 91 and 93 When the combination of RD cost and the success probability lies above the ray Eπ1 1 0 no RD is undertaken That is the combination of a low success probability or a high RD cost yields the decision that innovation is not undertaken even under monopoly conditions Figure 92 also shows that when the RD cost and probability combination lies between the curves Eπk 2 0 and Eπ11 0 only one firm engages in RD whereas if this combination lies below Eπk2 0 both firms undertake RD 922 Societys optimal RD level We now investigate what should be the number of firms that maximizes the societys welfare In general we should not expect that the equilib Page 226 Figure 92 RD race between two firms rium number of firms calculated in the previous subsection is necessarily optimal since the action of undertaking RD confers a negative externality on other firms engaging in the same RD race That is from a social welfare point of view increasing the number of firms engaging in RD will definitely increase the probability of discovery but will also increase the industrys aggregate RD cost associated with RD duplication Therefore without performing the actual calculation we find it hard to predict whether the equilibrium number of firms undertaking RD is below or exceeds the optimal number We denote by Eπsn the industrys expected profit when n firms undertake RD and associate the industrys expected profit with the welfare of the society When only one firm undertakes RD n 1 Thus when there is only one firm the social expected value of RD coincides with the firms expected profit from undertaking RD When there are two firms undertaking RD n 2 Page 227 Comparing Eπs1 with Eπs2 yields that Thus in terms of Figure 92 any combination above the Eπs1 Eπs2 curve is associated with the situation where the socially optimal number of firms engaged in RD is at most one Figure 92 is divided into four regions Region I The combination of high innovation cost and a low probability of discovery makes it unprofitable for even a single firm to undertake innovation It is obvious that if a single firm does not innovate it is not beneficial for a society to engage in this RD Region II These combinations of cost and discovery probability leave room for only one firm to undertake RD while still maintaining nonnegative expected profit Since cost is relatively high compared with the probability of discovery there are no social benefits from having a second firm engaging in RD Region III A relatively low innovation cost makes it profitable for a second firm to engage in RD However from the societys welfare point of view the cost of duplicating the RD effort 2I is larger than the societys benefits from the increase in the likelihood of getting the discovery as a result of having a second firm engage in RD This a case of market failure which occurs because firms do not take into account how their RD affect the profit of their rival firms Region IV These combinations involve a low innovation cost making it beneficial for both firms and the society to engage in the RD race despite the RD cost duplication Proposition 91 A market failure a condition in which it is socially desirable to have at most one firm engaging in RD but in equilibrium two firms engage in RD occurs only in Region III where the innovation cost I takes an intermediate value Formally In the literature patent races are generally analyzed in continuoustime models where the probability of discovery is a Poisson process that generates a constant probability of discovery at each point in time for a Page 228 given RD expenditure level see Loury 1979 Lee and Wilde 1980 and Reinganum 1989 for such modeling Fudenberg et al 1983 analyze an industry where the probability of discovery increases with the length of time in which the RD is conducted and derive the conditions for having one firm preempt others from racing toward a discovery see also Harris and Vickers 1985 923 Expected date of discovery Suppose that the race described in the previous subsection is repeated until one firm discovers the product Then what would be the expected date of discovery Before going to perform the calculations we need the following lemma The proof is given in an appendix Section 99 Lemma 91 Let δ satisfy 0 δ 1 Then Let Tn denote the uncertain date when at least one firm discovers the product given that n firms are engaged in RD for discovering the same product Also let ETn denote the expected date at which at least one firm discovers it A single firm When only one firm engages in RD n 1 the probability that T1 1 discovery occurs at the first date is α Next the probability that T1 2 discovery occurs at the second date is 1 αα That is the probability that the firm does not discover at the first date times the probability that it discovers at the second date Next the probability that the firm discovers at the third date is 1 α 2α Hence the expected date of discovery is given by Consequently if the probability of discovery is α ½ then ET1 2 and if α 13 then ET1 3 and so on Hence as expected an increase in the discovery probability a shortens the expected date of discovery Page 229 Two firms The probability that none of the firms discovers at a particular date is 1 α2 Hence the probability that at least one firm discovers at a particular date is 1 1 α2 α2 α Hence probT2 1 α2 α Next probT2 2 1 α2α2 α is the probability that none discovers at date 1 times the probability that at least one firm discovers at date 2 Therefore the expected date of discovery when two firms engage in RD is where the fourth equality sign follows from Lemma 91 Comparing 94 with 95 yields ET2 ET1 meaning that opening more independent research labs shortens the expected date of discovery 93 Cooperation in RD The antitrust legislation prohibits firms from engaging in activities that reduce competition and increase prices Any attempt at collusion is sufficient to provoke lawsuit against the cooperating firms However the antitrust legislation is less clear about how to handle cases where firms establish research joint ventures RJV or just decide jointly how much to invest in their separated labs The legal approach to RJV is addressed in the appendix Section 98 In this section we do not address problems such as how firms manage to implicitly or explicitly coordinate their research efforts and how the research information is shared by the participating firms see Combs 1993 and Gandal and Scotchmer 1993 Instead we analyze how firms determine their research efforts taking into consideration that they compete in the final goods market after the research is completed This problem has been the subject of many papers see Choi 1993 dAspremont and Jacquemin 1988 Kamien Muller and Zang 1992 Katz 1986 and Katz and Ordover 1990 In this section we analyze a twostage game in which at t 1 firms determine first noncooperatively and then cooperatively how much to invest in costreducing RD and at t 2 the firms are engaged in a Cournot quantity game in a market for a homogeneous product where the demand function is given by p 100 Q Page 230 The processinnovation RD technology We denote by xi the amount of RD undertaken by firm i i 1 2 and by ci x1 x2 the unit production cost of firm i which is assumed to be a function of the RD investment levels of both firms Formally let That is the unit production Cost of each firm declines with the RD of both firms where the parameter β measures the effect of firm js RD level on the unit production cost of firm i Formally Definition 92 We say that RD technologies exhibit positive spillover effects if β 0 That is if β 0 the RD of each firm reduces the unit cost of both firms For example spillover effects occur when some discoveries are made public during the innovation process some secrets are not kept Also this positive externality can emerge from the labs investing in infrastructure or from research institutes and universities that benefit all other firms as well see Jaffe 1986 for empirical evidence Assuming β 0 implies that RD exhibits only positive spillover effects However note that in some cases β can be negative if the RD of a firm also involves vandalism activities against competing firms such as radar jamming or spreading false information and computer viruses Finally to close the model we need to assume that RD is costly to firms Formally denote by TCi xi the cost for firm i of operating an RD lab at a research level of xi Assumption 92 Research labs operate under decreasing returns to scale Formally Assumption 92 implies that the cost per unit of RD increases with the size of the lab That is higher RD levels require proportionally higher costs of lab operation Note that this assumption heavily affects the results because if labs were to operate under increasing returns say by having to pay a high fixed cost for the construction of the lab firms would always benefit from operating Only a single lab that serves both firms when firms are allowed to cooperate in RD Subsection 931 calculates the firms profit maximizing RD levels when firms do not cooperate Subsection 932 calculates the RD levels that maximizes the firms joint profit when firms are allowed to coordinate their RD levels while still maintaining two separate labs Page 231 931 Noncooperative RD We look for a subgame perfect equilibrium Definition 210 where firms choose their RD expenditure levels in the first period and their output levels in the second periods We find this equilibrium by first solving for the Nash equilibrium in the second period and then working backwards we solve for the firstperiod RD levels The second period The secondperiod Cournot competition takes place after the cost reduction innovation is completed Hence the postinnovation c1 and c2 are treated as given Thus our Cournot analysis of section 61 on page 98 applies so if we recall 67 the Cournot profit levels are given by The first period In the first period each firm noncooperatively chooses its level of RD given the RD level of the rival firm That is we look for a Nash equilibrium Definition 24 on page 18 in RD levels Formally substituting 96 into 97 for a given level of xj firm i chooses xi to The firstorder condition yields Given that the payoff functions are symmetric between the two firms we look for a symmetric Nash equilibrium where where xnc is the common noncooperative equilibrium RD level invested by each firm when the firms do not cooperate Thus 932 Cooperative RD Under cooperative RD firms jointly choose RD levels that will maximize their joint profits knowing that in the second period they will compete in quantities Page 232 The firms seek to jointly choose x1 and x2 to where πi i 1 2 are given in 97 The firstorder conditions are given by The first term measures the marginal profitability of firm i from a small increase in its RD xi whereas the second term measures the marginal increase in firm js profit due to the spillover effect from an increase in is RD effort Hence Assuming that second order conditions for a maximum are satisfied the first order conditions yield the cooperative RD level We now compare the industrys RD and production levels under noncooperative RD and cooperative RD Proposition 92 1 Cooperation in RD increases firms profits 2 If the RD spillover effect is large then the cooperative RD levels are higher than the noncooperative RD levels Formally if then xc xnc In this case Qc Qnc 3 If the RD spillover effect is small then the cooperative RD levels are lower than the noncooperative RD levels Formally if then xc xnc In this case Qc Qnc Proof For part 1 clearly the firms could decide to set the RD at the noncooperative levels However if they set it means that their joint profit must increase Parts 2 and 3 follow from comparing 99 with 910 The quantity comparisons follow from the simple fact that in a Cournot market structure the aggregate quantity increases with a decline in unit production costs Page 233 The intuition behind parts 2 and 3 of Proposition 92 is as follows First note that under noncooperation each firm sets its RD level to reduce its own cost ignoring that fact that it reduces the cost of the other firm as well Now if β is high the spillover effect is intense then under cooperation the firms set RD levels higher than the noncooperative levels since under cooperation firms take into account the effect of their RD on their joint profits When the spillover effect is small the effect of each firm on the cost reduction of the other firm is small hence when firms do not cooperate each firm has a lot to gain from RD since under small spillover effects the RD intensifies the cost advantage of the firm that undertakes a higher level of RD Shaffer and Salant 1998 have pointed out some problems associated with the commonly used assumption that the two labs are engaged in an equal amount of RD They have shown that even though the aggregate RD cost of identical firms in a research joint venture would be the lowest if they invested equally to reduce subsequent production costs nonetheless members may enlarge their overall joint profit by instead signing agreements which mandate unequal investments If we apply their analysis to our simple example it turns out that unequal RD levels maximize joint profit if the spillover parameter β is sufficiently low 1 21 β2 or β 03 implying that we need to assume that β 03 in order to make the analysis of this section valid Finally in the present analysis the profit of firms must be higher under cooperation than under noncooperation since under cooperation in the first stage the firms can always invest at the noncooperative RD level and earn the same profit as under noncooperation However Fershtman and Gandal 1994 show that the profit of the firms may be lower under cooperation in a different game where firms compete in RD in the first period but collude in the second period This happens since depending on the secondperiod profitsharing rule each firm may overinvest in RD in order to negotiate a larger fraction of the cooperative profit in the second period 94 Patents A patent is a legal document granted by a government to an inventor giving the inventor the sole right to exploit the particular invention for a given number of years see an appendix Section 97 for a detailed analysis of patent law It is widely accepted that the patent system is useful for encouraging new product development and process innovation despite the market distortion it creates by granting temporary monopoly rights to new firms Thus the patent system is essential to growing economies Empirically it is very hard to measure the social value of Page 234 a patent since patented invention tend to be rapidly imitated or be patented around the patented innovation so the knowledge is diffused into many firms into other industries see Mansfield 1965 and into other countries One way to solve the problem how to measure the social value of a patented innovation is to count the number of times the innovation is cited in other patented innovations see Trajtenberg 1990 Formally the patent system has two social goals To provide firms with the incentives for producing knowhow and to make the new information concerning the new discoveries available to the public as fast as possible In other words society recognizes that information dispersion is a key factor in achieving progress and that public information reduces duplication of RD Note that the informationdissemination goal of the patent may somewhat contradict the pure interpretation of the patent law stating that a future innovation is patentable only if it does not infringe on earlier patented inventions That is on the one hand society desires to disclose the information behind the invention in order to enhance the research by other firms on the other hand other firms would not be able to patent a technology that infringes on older patents However providing the public with the information on patented technologies definitely reduces extra cost resulting from RD duplication in the sense that it prevents the wheel from being reinvented The reason why innovators need extra protection lies in the fact that knowhow is a very special entity compared with other products such as chairs cars and cheese knowhow is easy to duplicate and steal Once a firm makes its invention known to others other firms would immediately start with imitation followed by intense competition thereby reducing the price to unit cost With zero profits no firm would ever engage in RD and the economy would stagnate forever The goal of the patent system is to reward innovators The drawback of the system is that it creates a price distortion in the economy since those goods produced under patent protection will be priced differently from goods under no patent protection There are different kinds of patents such as patents given for a new product a new process or a substance and a design patent In order for an invention to be classified as worthy of a patent it has to satisfy three criteria it has to be novel nontrivial and useful In practice it is hard to measure whether an invention satisfies these criteria and therefore patents tend to be approved as long as they do not infringe on earlier patented innovations For a discussion of the legal side of the patent system and intellectual property see the appendix Section 97 This appendix discusses many important legal and economic aspects of patent protection In this section we discuss only one important and difficult aspect Page 235 of the patent system the duration of patent protection For example in the United States inventors are generally rewarded with seventeen years of patent protection and in Europe with around twenty years of protection Here we wish to investigate what factors affect a societys optimal duration of patents We now provide a simple method for calculating the optimal duration of a patent that was proposed in Nordhaus 1969 and Scherer 1972 As in section 93 consider a firm that is capable of undertaking a process innovation RD An investment of x in RD reduces the firms unit cost from c 0 to c x The cost of undertaking RD at level x is the same as in Assumption 92 We assume that the innovation is minor see Definition 91 so the innovating firm profitmaximizing price assuming that the unit cost of all competing firms remain c is p c Hence there will be no change in output as a result of the innovation Figure 93 illustrates the market before and after the process innovation reduces the unit cost of the innovating firm by x assuming a market demand given by p a Q where a c Since there is no change in Figure 93 Gains and losses due to patent protection price charged to the consumers the area M in Figure 93 measures the innovators gain in profit due to the innovation Assuming that the government sets the patent life for periods T 17 in the United States we see that the innovator enjoys a profit of M for T periods and zero profit from period T 1 and on The area DL in Figure 93 is the societys deadweight loss resulting from the monopoly power held by the patent holder for T periods That is in periods t 1 2 T the societys benefit from the innovation is Page 236 only the monopolys profits M assuming that the profits are distributed to consumers say via dividends In periods t T 1 T 2 after the patent expires all firms have access to the new technology and the equilibrium price falls to c x Hence after the patent expires the gain to the society is the sum of the areas M DL since the removal of the monopoly rights expands output and increases consumer surplus by DL It is clear from Figure 93 that Since the patent means monopoly rights for several periods we need to develop a dynamic model in order to determine the optimal patent duration Therefore let ρ 0 ρ 1 denote the discount factor Recall that the discount factor is how much a dollar next year is worth today In perfect markets the discount factor is also inversely related to the interest rate That is ρ 11 r where r is the market realinterest rate In what follows we consider a twostage game In the first stage the government sets the duration of the patent life T knowing how a firm would react and invest in costreducing RD In the second stage at t 1 the innovator takes the patent life as given and chooses his or her RD level Then during the periods t 1 T the innovator is protected by the patent rights and collects a monopoly profit for T periods 941 Innovators choice of RD level for a given duration of patents Denote by πx T the innovators present value of profits when the innovator chooses an RD level of x Then in the second stage the innovator takes the duration of patents T as given and chooses in period t 1 RD level x to That is the innovator chooses RD level x to maximize the present value of T years of earning monopoly profits minus the cost of RD We need the following Lemma The proof is given in an appendix Section 99 Lemma 92 Page 237 Hence by Lemma 92 and 911 912 can be written as implying that the innovators optimal RD level is Hence Proposition 93 1 The RD level increases with the duration of the patent Formally xIincreases with T 2 The RD level increases with an increase in the demand and decreases with an increase in the unit cost Formally xIincreases with an increase in a and decreases with an increase in c 3 The RD level increases with an increase in the discount factor p or a decrease in the interest rate The intuition behind Proposition 93 is as follows When the duration of patents increases the firm will be protected for a longer period and therefore will be selling more units over time Thus a higher RD level would correspond to a unitcost reduction for for a higher volume of production which would make the process innovation even more profitable The prediction of part 3 of Proposition 93 should remind you of your macroeconomics classes where the Keynesian and ISLM approaches assumed that investment increases when interest rates fall Here we obtain this result when the discount factor increases say due to a drop in the real interest rate the firms present value of discounted profits increases thereby making innovation more profitable 942 Societys optimal duration of patents We now turn to the first stage of the game where the government legislates the duration of the patent to maximize social welfare taking into account how the duration of patents affects the innovators RD level As represented in Figure 93 the societys welfare is CS0 M from the date the invention occurs and CS0 M DL from the date when the monopolys patent right expires Page 238 Formally the social planner calculates profitmaximizing RD 913 for the innovator and in period t 1 chooses an optimal patent duration T to Since and using 911 914 can be written as choosing T to maximize Thus the government acts as a leader since the innovator moves after the government sets the patent length T and the government moves first and chooses T knowing how the innovator is going to respond We denote by T the societys optimal duration of patents We are not going to actually perform this maximization problem in order to find T In general computer simulations can be used to find the welfaremaximizing T in case differentiation does not lead to an explicit solution or when the discrete nature of the problem ie T is a natural number does not allow us to differentiate at all However one conclusion is easy to find Proposition 94 The optimal patent life is finite Formally Proof It is sufficient to show that the welfare level under a oneperiod patent protection T 1 exceeds the welfare level under the infinite patent life When T 1 xI1 a c Hence by 915 When Hence by 915 Page 239 A comparison of 916 with 917 yields that To show that the last inequality in 918 holds for every 0 ρ 1 define Hence if we crossmultiply 918 it is sufficient to show that 2α2 α3 2α 1 for all 0 α 1 The latter holds if α2 α3 α2 2α 1 α2 α3 α 12 0 This inequality holds since each term is strictly positive The result obtained in Proposition 94 is important because it is often argued in the literature that innovators should be granted an infinite patent life The logic behind the infinitepatentlife argument is that in order to induce an innovator to undertake the optimal RD level the innovator should be rewarded with the entire profit stream from the innovation which could last forever That is without the infinite patent protection the innovator cannot capture all the rents from future sales associated with the innovation and hence will not innovate at the optimal level However Proposition 94 shows that the monopoly distortion associated with an infinitely lived monopoly is larger than the innovation distortion associated with an insufficient reward to the innovator Chou and Shy 1991 1993 have found that this result also holds for patents given for product innovation rather than for a process innovation in the present case Also Stigler 1968 provides an interesting calculation leading to an optimal patent life of seventeen years 95 Licensing an Innovation Licensing of technologies is common on both the national and the international scales Over 80 percent of the inventions granted patents are licensed to other firms where some are exclusively licensed and others are licensed to several manufacturers Given this observation we ask in this section why a firm that invested a substantial amount of resources in RD would find it profitable to license its technology to a competing firm that has not invested in RD Several answers to this questions are given in the literature on patent licensing and surveyed in Kamien 1992 We answer this question by considering the following example Consider the simple twofirm Cournot example illustrated in Figure 93 and suppose that firm 1 has invented a minor costreducing process indicated by a lower unit cost c1 c x where c is the unit cost of the noninnovating firm 2 c2 c Page 240 No licensing If firm 1 does not license its technology the firms play Cournot where in section 61 on page 98 we calculated that and That is firm 1 with the lower unit cost produces a higher amount and earns a higher profit than firm 2 Licensing Suppose that firm 1 negotiates with firm 2 for granting permission to firm 2 to use the less costly technology There can be several types of licensing For example there can be a fixedfee license a fee that is independent of the output produced by firm 2 or firm 1 can charge firm 2 with a perunit fee for every unit sold by firm 2 Consider a perunit fee case that is very common in the electronics and entertainment industries for example in which firm 2 buys the technology for producing at unit cost of c1 c2 and has to pay firm 1 the sum of φ for every unit it sells Although it is clear that the two firms have some surplus to divide between themselves when firm 2 buys the costsaving technology from firm 1 we take the simplest approach by assuming that firm 1 is a leader which offers firm 2 a takeitorleaveit contract to pay a perunit fee of φ In other words in the first stage firm 1 offers the technology to firm 2 for a perunit fee In the second stage firm 2 can either reject the offer or accept the offer and then choose how much to produce We now seek to find the profitmaximizing perunit of output fee φ that firm 1 charges firm 2 for its costreducing technology Clearly firm 1 sets That is firm 1 charges a perunit fee that is almost the size of the unit cost reduction associated with the licensed technology Therefore under this licensing contract the fee inclusive perunit cost facing firm 2 is now given by Hence in a Cournot equilibrium firm 2 would not change its quantity produced and therefore its profit level does not change In contrast the profit of firm 1 is now given by That is firm 1 gains all the surplus generated by the cost reduction in the production of firm 2 Therefore we can state the following proposition Proposition 95 1 In a Cournot environment licensing a costreducing innovation can increase the profit of all firms 2 In a Cournot environment welfare increases when firms license costreducing innovations Page 241 The last part of the proposition follows from the fact that in our example firms do not change their output levels and therefore the market price does not change Hence consumers welfare remains unchanged The profit of firm 1 increases however implying an aggregate welfare increase 96 Governments and International RD Races We observe that governments never completely leave RD to be performed by the free markets Governments intervention in RD starts with the establishment of mandatory school systems and universities and ends with direct subsidies to firms or industries In the developing countries the gross estimation of the domestic RD expenditure is around 3 to 35 percent of the GDP Out of that 30 to 60 percent is government financed In this section we analyze two examples in which international competition between firms located in different countries generates an incentive for each government to subsidize the RD for the firm located in its country Subsection 961 analyzes how a governmental subsidy to a domestic firm can secure the international dominance of the domestic firm in an international market for a new product Subsection 962 analyzes governmental subsidies to processinnovation RD 961 Subsidizing new product development Consider Krugmans 1986 illustration of how governments can enhance the international strategic position of the firms located in their countries Suppose that there are only two civilian aircraft manufacturers in the entire world and that the world consists of two countries the United States and the European Community Suppose that the US manufacturer is called Boeing and the European firm is called Airbus Each firm is considering developing the future superlarge passenger plane the megacarrier intended to transport six hundred passengers and having a flight range exceeding eighteen hours Suppose further that each firm has a binary choice develop and produce or dont develop and dont produce Table 91 demonstrates the profit levels of each firm under the four possible market outcomes Table 91 demonstrates what several civil aviation specialists frequently argue that given the high development costs a twofirm market is inconsistent with having positive profit levels That is in this kind of market there can be at most one firm earning strictly positive profit The Nash equilibrium see Definition 24 on page 18 for this game is given in the following proposition Page 242 AIRBUS Produce Dont Produce BOEING Produce 10 10 50 0 Dont Produce 0 50 0 0 Table 91 Profits of Boeing and Airbus under no govt intervention Proposition 96 In the BoeingAirbus game there emit exactly two Nash equilibria Produce Dont Produce and Dont Produce Produce Now suppose that the EC subsidizes Airbus by providing fifteen units of money for the development of a European megacarrier Table 92 illustrates the profit levels of the two aircraft manufacturers under the four possible outcomes AIRBUS Produce Dont Produce BOEING Produce 10 5 50 0 Dont Produce 0 65 0 0 Table 92 Profits of Boeing and Airbus under the EC subsidy In this case we can assert the following Proposition 97 Under the EC subsidy a unique Nash equilibrium is given by having Airbus play Produce and having Boeing play Dont Produce Thus by subsidizing product development a government can secure the world dominance of the domestic firm in a product having large development costs relative to the potential market size Although we have shown that the EC can guarantee its dominance in the megacarriers market by providing a subsidy to Airbus it is not clear that the welfare of the EC residents increases with such a policy since the EC residents will have to pay for this subsidy in one form or another 962 Subsidizing process innovation Following Brander and Spencer 1983 and 1985 consider two countries denoted by i 1 2 each of which has one firm producing a homogeneous product only for export to be sold in the world market The worlds Page 243 demand for the product is p a Q assume that the preinnovation unit cost of each firm is c where 0 c a Let xi denote the amount of RD sponsored by the government in country i We assume that when government i undertakes RD at level xi the unit production cost for the firm producing in country i is reduced to cxi i 1 2 As in Assumption 92 on page 230 we assume that the total cost to government i of engaging in RD at level xi is TCixi xi22 Since we assumed that the two firms play a Cournot quantity game in the world market for given RD levels x1 and x2 65 and 67 see section 61 on page 98 imply that the profit level of the firm located in country i is We denote by Wi the welfare of country i which is defined as the sum of the profit earned by firm i minus the RD cost Altogether each government i takes xj as given and chooses an RD level xi to maximize the welfare of its country That is government i solves The firstorder condition yields how the government of country i sets its RD level in response to the RD set is country j Thus Note that the countries bestresponse functions are strategic substitutes see Definition 72 on page 140 reflecting the fact that if one country increases its RD level the other reduces it Equation 919 shows that if country j does not subsidize its RD xj 0 then the government of country i sets a strictly positive RD level xi 4ac 0 Hence Proposition 98 If initially the world is characterized by no government intervention it is always beneficial for at least one country to subsidize RD That is the increase in profit from export sales associated with the costreducing RD dominates the cost of RD Solving 919 yields the unique symmetric Nash equilibrium RD levels given by Page 244 Proposition 99 In a Nash equilibrium of an RD game between two governments each government subsidizes the RD for the firm located in its country Also the equilibrium levels of the RD subsidies increase with a shift in the world demand a and decrease with the initial unitproduction cost c Thus when demand rises governments increase their RD subsidies since cost reduction is magnified by larger sales Finally the reader should not interpret this model as the ultimate argument for having governments subsidize RD of the exporting firms because this model does not explain why the government itself should perform the RD In other words why does the private sector not invest in RD given that the firms increase in profit can more than cover the RD cost Why cannot banks finance this innovation Also it is unlikely that governments possess all the information needed to decide which RD is profitable and which is not For arguments against protection see Baldwin 1967 For a comprehensive survey of strategic trade policy see Krugman 1986 The result obtained in this subsection has been mitigated in several papers First Dixit and Grossman 1986 have shown that in a general equilibrium model as compared with our partial equilibrium framework the incentive for protection becomes weaker Second Eaton and Grossman 1986 have shown that the choice of policy instrument for helping the domestic industry depends heavily on the assumed market structure Hence since governments never know exactly whether the market structure is Cournot or a different one the optimal policy may simply be not to intervene Third Gaudet and Salant 1991 show that the Brander and Spencer result is a special case because if one country has a large number of exporting firms and one has a small number of exporting firm the optimal policy for the government in the country with the large number of firms may be a tax instead of a subsidy that will induce some firms to exit 97 Appendix Patent Law A patent application is submitted to the Patent Office Then the Patent Office examines the application and does research to determine whether the claims made by the petitioner fulfill the criteria for granting a patent In many cases patents are denied by the Patent Office and the innovator resubmits the application During this time it often happens that other innovators apply for similar patents and in this case the question of who invented first has to be answered by the Patent Office After the patent is granted the patentee is given exclusive rights to Page 245 make use or sell the invention to the absolute exclusion of others In the United States the patent is granted for seventeen years and cannot be renewed 971 History of Patent Law The history of the patent system can be traced to medieval times in Europe when commerce became controlled by various groups and guilds The reader interested in more details is referred to Kaufer 1989 and Miller and Davis 1990 The earlier patents issued by the Crown in England were a method used by the monarch to control various sectors in return for some benefits That is early patent rights were not as concerned with inventions as with the protection of the monarchy itself In 1623 the Statute of Monopoly ended the period of unrestricted granting of monopolies by the Crown In fact the development of patent law was needed to secure monopoly rights for special reasons such as to reward the innovators rather than for the unrestricted granting of monopoly rights In 1624 England passed a statute to regularize previously arbitrary letters of patents issued by the Crown The life of a patent was set at fourteen years because fourteen is two times seven and seven years was the normal length of an apprenticeship the time needed to train a professional say a doctor Then the patent could be extended for seven additional years reaching a maximum number of twentyone years of patent protection It is possible that the current US system of seventeen years represents a compromise between fourteen and twentyone years In the New World the colonies began granting patents the colonists recognized that society could benefit from rewarding the innovators All this led to the statement in the US Constitution that The Congress shall have the powerTo promote the progress of science and useful arts by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries Then in 1836 the US Patent Office was given the authority to examine proposed inventions and to determine whether they meet the criteria of the Patent Statute In what follows we will refer to the Patent Act of 1952 as the Patent Law 972 Types of Patents A patent can be granted for products processes plants and design However any invention related to abstract ideas is not patentable For example the first person to prove Lemma 91 on page 228 or any other Page 246 lemma in this book was not entitled to a patent right since this invention is classified as an abstract idea or a mathematical formula However note that applications for abstract ideas of theories may be patentable 973 Criteria for granting a patent In order for an invention to be entitled to a patent it has to satisfy three requirements novelty nonobviousness and usefulness According to the patent law novelty refers to the lack of prior domestic or foreign patenting publication use or sale Nonobviousness refers to the requirement that the invention must demonstrate some advance over prior art so that the ordinary mechanic skilled in prior art would not have been capable of making this advance The purpose of the usefulness or utility requirement is to prevent patenting inventions that are based only on ingenuity and novelty but do not serve any purpose This requirement also intends to steer the RD towards inventing welfareincreasing inventions rather than useless ones 974 First to invent versus first to file The US patent law differs from those of other countries in one major respectthe priority assignment given to one of several agents filing for the same patent The general rule in the United States is that the innovator is the one who conceived first However one exception prevails the case in which a second innovator reduces the invention into practice and the first innovator did not exercise continuous diligence Thus an innovator who is the first to conceive the innovation and the first to reduce it to practice has a definite priority in getting the patent The US system is referred to as the firsttoinvent system which is not exercised by other countries The EC and Japan use a different priority system referred to as the firsttofile system Obviously the firsttofile system is easier to enforce Problems arise nowadays when claiming a priority over international patents since an invention could be recognized by one patent system but not the other 975 Copyrights Copyright gives an exclusive right to the copyright owner to reproduce the work and its derivatives in the form of copying or recording and are given on the basis of pure originality which refers to the act of authorship or artistic creativity and not necessarily on novelty The duration of the copyright ownership extends to the authors lifetime plus fifty years To obtain a copyright ownership the author or the artist must demon Page 247 strate that he or she has contributed something to the final production or a reproduction Thus a reproduction of a book in modem style or with new decorations may be eligible for copyright protection because the author or artist has contributed something that did not exist in the earlier version The Copyright Act also allows computer programs and sound recordings to receive copyright protection Finally the law permits reproduction of various works mainly for noncommercial purposes such as education 98 Appendix The Legal Approach to RD Joint Ventures Two major questions are faced by the regulators regarding cooperative RD First whether the act of joining together itself reduces competition thereby violating antitrust laws More precisely should RD joint ventures be considered as procompetitive or anticompetitive in the products market Second even if RD joint ventures are anticompetitive are there efficiency gains associated with joint RD that dominate the welfare loss resulting from anticompetitive behavior in the finalgood market Clearly unless the RD joint ventures offer gains in efficiency associated with more productive and less costly RD there is no reason to permit it For this reason antitrust cases brought against firms cooperating in RD are judged by the rule of reason rather than by the per se rule The following discussion of the legal approach to cooperative RD is based on Brodley 1990 and Jorde and Teece 1990 The US legal system seems to be less supportive of RD joint ventures than the EC and Japan According to the Clayton Act allegations that firms use price fixing permit suing for treble damages Therefore there is a question of whether cooperation in RD can open a channel of communication among firms to explicitly or implicitly collude on prices Despite these suspicions Congress has recognized the potential benefits associated with cooperative RD and in 1984 enacted the National Cooperative Research Act NCRA which states that joint RD ventures must not be held illegal per se The NCRA established a registration procedure for joint RD ventures The firms that do follow the registration procedure are immune from paying treble damages on any antitrust violation Instead the maximum penalty for registered firms is limited to damages interest and costs In sum the US law attempts to distinguish between joint RD and joint commercialization decisions by cooperating firms The former is legal and the latter is illegal The reader should note that sometimes Page 248 it is hard to distinguish between the two processes since the decision to commercialize an invention can be viewed as the last step of the RD process That is it is possible that one firm has a comparative advantage in theoretical product development while the other has one in making an innovation marketable In this case society may benefit from the formation of a joint venture despite the fact that joint commercialization may result in higher prices than those that obtain under pure competition since otherwise there might be no product at all This approach is more common in Japan where commercialization is an integral part of the RD process 99 Mathematical Appendix Proof of Lemma 91 First recall the high school identity given by Next Proof of Lemma 92 Using the high school identity given at the beginning of this appendix section we have it that 910 Exercises 1 Consider the classification of process RD given in section 91 Suppose that the aggregate inversedemand function is given by p a Q and Page 249 that initially all the firms have identical unit costs measured by c0 where c0 a 2c0 Suppose that one and only one of the firms is able to reduce its unit cost to c1 2c0 a Using Definition 91 infer whether this process innovation is considered to be minor or major 2 Consider a threefirm version of the patentrace model studied in section 92 Suppose that each one of the three firms is capable of developing a product Let V denote the monetary value of the patent associated with the new product Each firm can construct a research lab provided that it invests I in the lab Assume that if a firm constructs a lab it has a probability of α 12 of discovering the product If only one firm discovers the product it will earn a profit equal to the full value of the patent ie V If only two firms discover then each will earn V2 and if all three discover then each will earn V3 Answer the following questions a Assuming that I 1 calculate the minimal value of V that ensures that each firm will invest in constructing a lab b Suppose now that firm 3 went out of business and that a foreign firm purchased the two remaining firms Calculate the minimal value of V that would induce the foreign owner of the two firms to run the two separate research labs instead of operating only one lab 3 Consider the calculations of the expected time of discovery described in subsection 923 Suppose that n firms are engaged in RD where the probability of discovery by each firm at each date is α 0 α 1 Answer the following questions a What is the probability that none of the firms discovers at a particular date b What is the probability that at least one firm discovers at a particular date c Calculate the expected date of discovery 4 Consider the BoeingAirbus game described in Table 91 on page 242 a Calculate the minimal subsidy to Airbus that will ensure that Airbus will develop the megacarrier Explain b Suppose that the EC provides Airbus with fifteen units of money as a subsidy Which subsidy by the US government to Boeing would guarantee that Boeing will develop this megacarrier c Suppose that the EC provides Airbus with fifteen units of money as a subsidy Is there any level of subsidy given by the US government that would deter Airbus from developing this airplane d From your answer to the previous question conclude whether the world benefits by having both governments subsidizing their own aircraft manufacturing firms Explain Page 250 911 References Baldwin R 1967 The Case Against InfantIndustry Tariff Protection Journal of Political Economy 77 295305 Brander J and B Spencer 1983 International RD Rivalry and Industrial Strategy Review of Economic Studies 50 707722 Brander J and B Spencer 1985 Export Subsidies and International Market Share Rivalry Journal of International Economics 18 83100 Brodley J 1990 Antitrust Law and Innovation Cooperation Journal of Economic Perspectives 4 97112 Choi J 1993 Cooperative RD with Product Market Competition International Journal of Industrial Organization 11 553571 Chou C and O Shy 1991 New Product Development and the Optimal Duration of Patents Southern Economic Journal 57 811821 Chou C and O Shy 1993 The CrowdingOut Effects of Long Duration of Patents RAND Journal of Economics 24 304312 Combs K 1993 The Role of Information Sharing in Cooperative Research and Development International Journal of Industrial Organization 11 535551 dAspremont C and A Jacquemin 1988 Cooperative and Noncooperative RD in Duopoly with Spillovers American Economic Review 78 11331137 Dosi G 1988 Sources Procedures and Microeconomic Effects of Innovation Journal of Economic Literature 26 11201171 Dixit A and G Grossman 1986 Targeted Export Promotion With Several Oligopolistic Industries Journal of International Economics 21 233249 Eaton J and G Grossman 1986 Optimal Trade and Industrial Policy under Oligopoly Quarterly Journal of Economics 2 383406 Fershtman C and N Gandal 1994 Disadvantageous Semicollusion International Journal of Industrial Organization 12 141154 Freeman C 1982 The Economics of Industrial Innovation 2nd ed Cambridge Mass MIT Press Fudenberg D R Gilbert J Stiglitz and J Tirole 1983 Preemption Leapfrogging and Competition in Patent Races European Economic Review 22 331 Gandal N and S Scotchmer 1993 Coordinating Research Through Research Joint Ventures Journal of Public Economics 51 173193 Gaudet G and S Salant 1991 Increasing the Profits of a Subset of Firms in Oligopoly Models with Strategic Substitutes American Economic Review 81 658665 Page 251 Harris C and J Vickers 1985 Perfect Equilibrium in a Model of Race Review of Economic Studies 52 193209 Jaffe A 1986 Technological Opportunity and Spillovers of RD Evidence from Firms Patents Profits and Market Value American Economic Review 76 9841001 Jorde M and D Teece 1990 Innovation and Cooperation Implications for Competition and Antitrust Journal of Economic Perspectives 4 7596 Kamien M 1992 Patent Licensing In Handbook of Game Theory edited by R Aumann and S Hart Amsterdam NorthHolland Kamien M E Muller and I Zang 1992 Research Joint Ventures and RD Cartel American Economic Review 82 12931306 Katz M 1986 An Analysis of Cooperative Research and Development Rand Journal of Economics 17 527543 Katz M and J Ordover 1990 RD Cooperation and Competition Brookings Papers on Economic Activity Microeconomics 137203 Kaufer E 1989 The Economics of the Patent System New York Hardwood Academic Publishers Krugman P 1986 Strategic Trade Policy and the New International Economics Cambridge Mass MIT Press Lee T and L Wilde 1980 Market Structure and Innovation A Reformulation Quarterly Journal of Economics 94 429436 Loury G 1979 Market Structure and Innovation Quarterly Journal of Economics 93 395410 Mansfield E 1965 Rates of Return from Industrial RD American Economic Review Papers and Proceedings 55 741766 Mokyr J 1990 The Lever of Riches Technological Creativity and Economic Progress Oxford Oxford University Press Miller A and M Davis 1990 Intellectual Property Patents Trademarks and Copyright in a Nutshell 2nd ed St Paul Minn West Publishing Nordhaus W 1969 Invention Growth and Welfare A Theoretical Treatment of Technological Change Cambridge Mass MIT Press Reinganum J 1989 The Timing of Innovation Research Development and Diffusion In Handbook of Industrial Organization edited by R Schmalensee and R Willig Amsterdam North Holland Rosenberg N 1994 Exploring the Black Box Cambridge Cambridge University Press Scherer F M 1972 Nordhaus Theory of Optimal Patent Life A Geometric Reinterpretation American Economic Review 62422427 Shaffer G and S Salant 1998 Optimal Asymmetric Strategies in Research Joint Ventures International Journal of Industrial Organization 16 195208 Page 252 Stigler G 1968 The Organization of the Industry Homewood Ill Richard D Irwin Trajtenberg M 1990 A Penny for Your Quotes Patent Citations and the Value of Innovation Rand Journal of Economics 21 172187 Page 253 Chapter 10 The Economics of Compatibility and Standards Standards are always out of date That is what makes them standards Alan Bennet Forty Years On 1969 Perhaps the most easily observed phenomenon is that people do not live alone People and all other animals tend to live in groups called villages towns cities or countries since they benefit from interacting with other people In addition to the pure social observation that people just enjoy being around other people the benefits of being and working together can be explained as follows Production Most production processes involve teams or groups of people using other complementary intermediate products such as machinery and computers Therefore for the production to be efficient machinery computers and all other equipment supporting workers must be designed in a way that a different workers would be able to use the same equipment and b the output generated by a certain machine would be able to be used by another worker operating a different machine Consumption People enjoy consuming goods that are also used by other people They like to watch the same movies to exchange books and to listen to music of the same composers People observe what others buy and try to match their consumption with that of their neighbors Page 254 Thus we can conclude that product or brand compatibility affects both the productivity of workers and the welfare of consumers In what follows we start with some descriptive definitions Later on in the chapter we shall give more precise definitions Definition 101 1 Brands of products are said to be compatible if they can work together in the sense that the output of one brand can be operated or used by other brands In this case we say that the brands operate on the same standard 2 Brands are said to be downward compatible if a newer model is compatible with an older model but not necessarily the other way around 3 Consumers preferences are said to exhibit network externalities if the utility of each consumer increases with the number of other consumers purchasing the same brand Examples for compatibility include products such as video and audio equipment records and tapes languages railroad gauges power supply computer operating systems computer software communication equipment the phone system fax and telex machines cellular and radio phones keyboards QWERTY versus DVORAK and banks and automatic teller machines ATMs More precisely video tapes operate on various different standards such as VHS Beta and different sizes such as 8mm and VHS size Music is recorded on LPs longplay records compact cassettes and Compact Disks CD Cellular phones which use airwaves instead of cables are used in two different standards analog or digital The commonly used QWERTY the first six letters on the upper row of the keyboard English keyboard was designed so that it slows the typist since fast typing is technically impossible on mechanical typewriters The newer DVORAK system allows faster typing however people were reluctant to switch to it see David 1985 Compatibility of automatic teller machines comes into effect when a customer carrying a bank card issued by one bank can withdraw cash from a machine servicing the clients holding a card issued by another bank In fact in Israel all the banks collude in the sense that any banks card can be used on all teller machines We will show later in this chapter that this behavior is indeed profitable to banks Finally extensions to the sevenbit ASCII code the most widely used as a standard for saving and transmitting computer files to eightbit for the purpose of increasing the number of characters from 27 to 28 Page 255 yielded several incompatible standards offered by MSDOS Macintosh and other computers Downward compatibility is commonly observed in the software industry where a newer version can read output files generated by the old version but in many cases the older version cannot input files generated by the newer version An example for preferences exhibiting network externalities include all communication equipment That is it is unlikely that a person would purchase a phone knowing that nobody else uses it To illustrate the significance of the choice of standards on the profits of firms in a certain industry Table 101 demonstrates a twofirm industry producing a product that can operate on two standards standard a and standard β Table 101 demonstrates a normal form game FIRM B Standard Standard β FIRM B Standard α a b c d Standard β d c b a Table 101 Standardization game where each firm can choose to construct its product to operate on standard a or standard β The profits levels of the two firms are given by the nonnegative parameters a b c and d where the profit of each firm is affected by the standard choices of the two firms We look for the Nash equilibria for this game Definition 24 on page 18 Proposition 101 1 If a b maxc d then the industry produces on a single standard that is α α and β β are Nash equilibria 2 If c d maxa b then the industry produces on two different standards that is α β and β α are Nash equilibria Part 1 of Proposition 101 resembles the Battle of the Sexes game see Table 22 on page 17 where the profit levels are high when the firms produce compatible brands on the same standard Industrywide compatibility is observed in the banking industry ATM machines and in many electronic appliances industries Part 2 of the proposition demonstrates a polar case where the firms can increase their profit by differentiating their brands and hence by constructing them to operate on different standards Examples for this behavior include the computer industrys producing computer brands operating on different operating Page 256 systems and automobiles that are produced with modelspecific parts Thus in this chapter we investigate firms incentives to standardize and the effects of their choices on consumers welfare There is a substantial amount of literature on compatibility issues For a comprehensive discussion on the nature of standards see Kindleberger 1993 For literature surveys see Farrell and Saloner 1987 David and Greenstein 1990 and Gabel 1991 Gandal 1994 provides some empirical evidence for the existence of network externalities in the computer software industry Our discussion of the economics of standardization is divided into three approaches Section 101 Network Externalities analyzes an industry where consumer preferences exhibit network externalities Section 102 Supporting Services shows that peoples tendency to use products that are identical or compatible to the products purchased by others need not be explained by assuming that consumers preferences exhibit network externalities That is it is possible that people will end up using compatible products even if their welfare is not directly affected by the consumption choice of other people Section 103 Components analyzes interface compatibility of components that are to be combined into a single usable system Two applications of these theories are not discussed in this chapter First Conner and Rumelt 1991 provides an application of network externalities to explain why software firms do not always protect the software against copying Second an application is discussed in section 171 where we show that when the choice of restaurants depends on the choice of other consumers a restaurant may refrain from raising its prices even when it faces a demand that exceeds its seating capacity 101 The Network Externalities Approach In this section we present the basic networkexternality model where consumers valuation of a brand increases with the number of other consumers using the same brand 1011 The interdependent demand for communication services One of the first attempts to model the aggregate demand for communication services is given in Rohlfs 1974 The demand for phone services Our point of departure is that the utility that a subscriber derives from a communication service increases as others join the system Consider Page 257 a group of a continuum of potential phone users indexed by x on the unit interval 01 Unlike the study of the Hotelling location model of subsection 731 in which we interpreted consumers indexed by a high x as consumers oriented toward brand B and consumers indexed by a low z as consumers oriented toward brand A here since we have only one type of service we interpret consumers indexed by a low x as those who love to subscribe to a phone system high willingness to pay and consumers indexed by a high x as those who have less desire for subscribing to a phone system low willingness to pay We denote by n the total number of consumers who actually subscribe to the phone system and by p the price of subscribing to the phone system Altogether we define the utility of a consumer indexed by x as Thus the utility of each subscriber exhibits network externalities since it increases with n the number of consumers subscribing to the phone system We now derive the consumers aggregate demand for phone services We first look at a particular consumer denoted by who is at a given price p indifferent to the alternatives of subscribing to the phone system and not subscribing In view of 101 the indifferent consumer is found by Since the number of consumers is given by we have it that which is drawn in Figure 101 The price p0 in Figure 101 intersects twice the flipped Ushaped curve at points and The interpretation for the two intersection points is that for a given price p0 there can be two levels of demand a low level measured by that is associated with a small number of subscribers hence by 101 with a low valuation by each subscriber and therefore with a small number of users and so forth In contrast at the given price p0 there can be a high demand measured by hence a high valuation by each subscriber and therefore a large number of subscribers and so forth However only point is a stable demand equilibrium since at the intersection point a small increase in the number of subscribers would make the phone subscription more desirable thereby causing all the consumers indexed on to subscribe Page 258 Figure 101 Deriving the demand for telecommunication services The point is defined in the literature as the critical mass for a given price p0 to indicate that at a given price any increase in the number of subscribers would shift the demand number of subscribers to the point The problem of the monopoly phone company Now suppose that there is only one monopoly firm providing phone services and suppose that the marginal cost of adding a subscriber is negligible after the PTT Public Telephone and Telegraph company has already wired all the houses We now ask what price maximizes the PTTs profit equals revenue in our case To solve this problem we formulate the PTTs profitmaximization problem which is to choose that solves The profit function 103 is drawn in Figure 102 The first and secondorder conditions for 103 are given by Now equation 104 and Figure 102 completely describe how the profit level is affected by changing the number of subscribers Clearly the profit is zero when there are no subscribers The profit is Page 259 Figure 102 The PTT profit function in the presence of network externalities also zero when the entire population subscribes since in order to have the entire population subscribing the PTT should set the price to zero The firstorder condition shows that and are extremum points In addition the second order condition shows that the second derivative is negative for implying that is a local maximum point Since the firstorder condition is positive for all it must be that is a global maximum point Hence Proposition 102 A monopoly phone companys profitmaximizing subscription price is set such that the number of subscribers exceeds half of the consumer population but is less than the entire population 1012 The standardizationvariety tradeoff In the previous subsection we confined the analysis to a single service In this subsection we develop a different model in which we assume that there are two brands of the product and heterogeneous consumers in the sense that each consumer prefers one brand over the other There are two firms each producing a different brand brand A and brand B We assume a continuum of consumers normalize the population size to 1 and assume that a 0 a 1 consumers prefer brand A over brand B whereas b 0 b 1 consumers prefer brand B over brand A where a b 1 The Farrell and Saloner 1986 model assumes that the utility of each consumer type increases with the number of consumers buying the same brand However if a consumer purchases the less desired brand his utility falls by δ 0 Formally the utility functions of types A and Page 260 B consumers are given by where xA denotes the number of consumers purchasing brand A and xB denotes the number of consumers purchasing brand B xAxB 1 The parameter δ also reflects the extra amount of money that a consumer is willing to pay to get his or her ideal brand Definition 102 1 If xA 1 and xb 0 we say that the product is standardized on A 2 If xA 0 and xB 1 we say that the product is standardized on B 3 If xA 0 and xB 0 we say that the product is produced with incompatible standards 4 An allocation of buyers between brands xA and xB is called an equilibrium if no single buyer would benefit from switching to the competing brand given that all other consumers do not switch from their adopted brand Equilibrium adoption of brands We first seek necessary conditions for a single standard to be an equilibrium Observe that in the following analysis since we assume a continuum of consumers each consumer is negligible in the sense that if a single consumer switches from buying brand A to buying brand B then it will not affect the aggregate the number of A and B users measured by xA and xB Now if the industry is standardized on A xA 1 then it must be that type B consumers would not benefit from switching from A to B implying that 1 δ 0 That is a consumer prefers to consume the same brand as the others rather than consuming alone his or her most preferred brand ie if the network effect dominates the ideal good effect Therefore Proposition 103 1 If δ 1 then two equilibria exist one in which A is the standard xA 1 and one in which B is the standard xB 1 2 If δ 1 no singlestandard equilibrium exists Page 261 We now investigate under what conditions the industry will produce two incompatible brands that is under what conditions xA a and xB b is an equilibrium In this equilibrium a type A consumer would not switch to B if a b δ Since b 1 a we have it that Similarly type B would not switch if Hence Proposition 104 If the number of each type of consumers is sufficiently large then there exists a twostandard equilibrium Formally if then xA a xB b is an equilibrium Figure 103 illustrates the parameter range for which the twostandard equilibrium exists As the utility loss from consuming the less preferred Figure 103 Twostandard incompatibility equilibrium brand parameter δ increases the parameter range for which incompatibility is an equilibrium increases That is if a twostandard equilibrium always exists Efficiency of brand adoption We define the economys social welfare function as the sum of consumers utilities Formally let In view of the three possible outcomes described above we have it that Page 262 Comparing these social welfare levels yields Proposition 105 If there are more consumers oriented toward brand A than there are consumers oriented toward brand B a b then standardization on A is socially preferred to standardization on B We now ask under what condition the incompatibility equilibrium outcome is socially preferred to a singlebrand standardization It follows from 106 that incompatibility is preferred over standardization on A if a2 b2 a b bδ 1 bδ or Using the fact that b 1a we see that this last condition is equivalent to δ2a or Similarly incompatibility is socially preferred over standardization on B if However these conditions cannot both hold if δ 1 since in this case Hence Proposition 106 1 If the network preference effect is strong relative to the disutility from consuming the less preferred brand δ 1 then the incompatibility equilibrium is socially inefficient 2 If δ 1 incompatibility is socially optimal if and Is there a market failure We first ask whether standardization on a singlebrand equilibrium may not be socially desirable Proposition 103 shows that as long as δ 1 there are two equilibria in which the industry produces on a single standard However 106 implies that if there are more consumers oriented toward A standardization on A socially dominates standardization on B Hence Proposition 107 An equilibrium in which the industry standardizes on the less socially preferred brand exists However note that in this case there is also a good equilibrium where the industrys standard is the more popular brand so one can assume that with a minor coordination consumers can choose the socially preferred standard How can it happen that an industry specializes on the wrong brand Consider a dynamic scenario which is not analyzed in this section such that a b and brand B exists in the market before brand A attempts to enter the market In this case the firm producing brand A will not be able to enter the market In the literature this situation is generally described as a case where the existence of an installed base brand B has prevented the emergence of the more popular brand A Page 263 We now seek to investigate whether a market failure can occur under the incompatibility equilibrium Let us take an example a b 05 and δ 06 Proposition 104 implies that incompatibility is an equilibrium since 12 1 062 02 However since δ 06 1 Proposition 106 implies that incompatibility is inefficient Hence Proposition 108 An equilibrium in which the industry produces two incompatible brands need not be socially efficient Finally the opposite of Proposition 108 holds Proposition 109 If incompatibility xA a and xB b is efficient then the incompatibility equilibrium exists and is unique Proof If incompatibility is efficient then part 1 of Proposition 106 implies that δ 1 Since a 0 and b 0 Proposition 104 implies that incompatibility is an equilibrium Also Proposition 103 implies that an equilibrium where an industry is standardized on a single standard does not exist 102 The Supporting Services Approach The analysis of the previous section was based on the assumption that consumers value for a product increases when other consumers purchase a compatible or an identical brand However despite the fact that the networkexternalities assumption is intuitive and appealing for modeling products such as telecommunication systems where the utility of each consumer is directly related to the network size the models themselves do not explain why people behave this way So the remaining question is whether network effects can prevail even without assuming that consumers preferences exhibit network externalities We therefore turn now to models describing consumers who do not derive satisfaction from the consumption of other consumers Instead consumers gain satisfaction from the product itself and the variety of brandspecific complementary products that we call supporting services The literature utilizing this approach includes Chou and Shy 1990 1993 and 1996 and Church and Gandal 1992ab 1993 In many instances supporting services are incompatible across brands For examples most software packages are designed to operate on one operating system such as UNIX DOS Macintosh OS etc and do not operate on the other operating systems Videotapes recorded on the NTSC television system used in North America and Japan cannot be played in Europe or in the Middle East where the dominant television standard is PAL For a discussion of the newly emerging highdefinition television standards see Farrell and Shapiro 1992 and the references therein Page 264 1021 Network effects without network externalities Consider consumers who can freely choose between two computer brands named brand A short for Artichoke computers and brand B short for Banana computers Each consumer is endowed with Y dollars to be spent on one unit of hardware and the variety of software written for the specific hardware purchased We denote by pi the price of computer brand i i A B Hence given a total budget of Y a consumer purchasing brand i spends on is specific software We denote by Ni the total number of software packages that can be run on an i machine The utility of a consumer purchasing system i is defined as an increasing function of the number of software packages compatible with machine i i A B Consumers are uniformly indexed by δ on the interval 01 according to their relative preference towards computer brand B We define the utility of a consumer type δ as Thus the utility function 107 describes preferences exhibiting love for variety of software That is a consumers preferences toward a specific brand are affected by a fixed parameter δ or 1 δ and by the number of software packages available for each brand NA and NB Figure 104 illustrates how consumers are distributed according to their preferences toward the two brands Figure 104 Consumers distribution of tastes The consumer who is indifferent to the choice between system A and system B is denoted by which is found from 107 by solving Thus in equilibrium a consumer indexed by is an Auser whereas a consumer indexed by is a Buser The total number of Ausers is denoted by and the total number of B users is given by Altogether Page 265 Hence Proposition 1010 The brand with the higher market share is supported by a larger variety of software Formally if and only if Proposition 1010 confirms widely observed phenomena for example the Intelbased machines PCs have the largest market share and are supported by the largest variety of software compared to machines based on other chips The software industry We have not yet discussed how the variety number of each brandspecific software is being determined in each software industry Instead of fully modeling the software industry we conjecture that the number of software packages supporting each machine should be proportional to the aggregate amount of money spent on each type of software We therefore make the following assumption Assumption 101 The number of software packages variety supporting each brand is proportional to the aggregate expenditure of the consumers purchasing the brandspecific software Formally Substituting into 109 yields Network effects The following proposition part 4 in particular demonstrates how network effects can prevail without assuming network externalities Proposition 1011 An increase in the price of hardware A pA will 1 decrease the number of Ausers δA decreases 2 increase the number of Busers δB increases 3 decrease the variety of software written for the A machine NA decreases and increase the variety of Bsoftware NB increases and Page 266 4 decrease the welfare of Ausers and increase the welfare of Busers Proof Part 1 follows from 1010 since Part 2 immediately follows since δB 1 δA Part 3 follows from Assumption 101 since as decreases and pA increases it is implied that NA must decrease while NB must increase Part 4 follows from 107 since a decrease in NA decreases the utility of an Auser whereas an increase in NB increases the utility of a Buser When pA increases Assumption 101 implies that two factors exist that cause a reduction in the variety of Asoftware First the direct effect Y pA decreases that is Ausers spend more on hardware and therefore less on software and second the indirect effect via a reduction in the number of Ausers decreases Assumption 101 also implies that NB increases since there are more Busers Part 3 of Proposition 1011 demonstrates the network effect generated by an increase in hardware price pA on the welfare of Busers as follows That is a decrease in the number of Ausers causes an increase in the number of Busers which in turn increases the variety of Bsoftware which increases the welfare and number of Busers and so on 1022 Partial compatibility Note that 100 percent compatibility is never observed For example you have probably noticed that sometimes you fail to transmit a fax to a remote fax machine because the other machine does not fully respond to all standards You have probably also noticed that some record and tape players are not rotating at the same speed Also even when the manufacturer asserts that his computer say is DOS compatible there are always some packages of software that can operate on one machine but refuse to operate on another In that sense 100 percent compatibility is actually never observed Perhaps the main advantage of using the supportingservices approach to model network behavior is that it allows an easy interpretation for modeling the concept of partial compatibility Definition 103 A computer brand i is said to be partially compatible with a pi degree of compatibility with computer brand j if a fraction pi of the total software written specifically for brand j can also be run on computer brand i It should be pointed out that Definition 103 does not imply that compatibility is a symmetric relation In other words it is possible that Page 267 a computer of a certain brand is designed to be able to read software developed for rival machines but the rival machines are not designed to read software not specifically designed for them In the extreme case in which pi 1 but pj 0 machine i can read j software but machine j cannot read i software we say that the machines are oneway compatible The number of software packages written specifically for machine i is denoted by ni i AB The main feature of this model is that the machines can be partially compatible in the sense that in addition to its own software each machine can also run a selected number of software packages written for its rival machine That is pi measures the proportion of machine j software that can be run on an i machine i j A B and Therefore the total number of software packages available to an imachine user is equal to We will not develop the complete model The complete computer and software industry equilibrium is developed in Chou and Shy 1993 Instead in what follows we merely illustrate the main insights of this model Suppose that the software industry produces a positive variety of both types of software That is nA 0 and nB 0 Now for the sake of illustration let NA and NB be kept constants Figure 105 shows the equilibrium nA and nB levels associated with the given NA and NB Figure 105 Equilibrium variety of brandspecific software Page 268 The line NA shows the combinations of brandspecific software nA and the rival brandspecific software nB associated with a constant level NA of Ausable software available to Ausers for a given level of compatibility ρA Similarly the line NB shows all the nA and nB combinations associated with a constant level of Busable software NB The point is the equilibrium variety of software written specifically for A and B machines Now suppose that the producer of computer A makes its machine more compatible with B software ie ρA increases Hence the line NA tilts to the left because in order to keep the number of A usable software at a constant level there is less need for Aspecific software since Ausers can use more of Bsoftware Therefore the new softwarevariety equilibrium is now given at the point in Figure 105 Consequently Proposition 1012 When there are two software industries each producing brandspecific software an increase in the degree of compatibility of the Amachine with the software written for the Bmachine 1 will reduce the variety of software specifically written for the Amachine nA decreases 2 will increase the variety of software specifically written for the Bmachine nB increases and 3 will reduce the total variety of software available to Ausers and will increase the total variety of software available to Busers NA decreases and NB increases The last part of the Proposition is proved in Chou and Shy 1993 The significance of the proposition which was actually known to many computer makers a long time before it was known to economists is that it shows that a computer manufacturer may refrain from making its machine more compatible with the software supporting the rival machine because compatibility with the rival machines software will induce software writers to write more software for the rival machine since part of it is usable for both machines thereby making the rival machine more attractive to consumers This result explains why computer manufacturers may choose different operating systems for their machines It should be pointed out that there could be reasons other than the one in Proposition 1012 for why firms make their brand less compatible with other brands For example in subsection 1222 we show other cases in which firms choose to differentiate themselves from other firms by producing products of different quality Page 269 103 The Components Approach In the previous sections we introduced two approaches to the economics of networks a the networkexternality approach where a consumers valuation of a certain brand is affected by the number of consumers purchasing a similar or an identical brand and b the supportingservices approach where a consumers valuation of a brand is affected by the number of supporting services supporting software supporting the specific brand The components approach discussed in this section is similar to the supporting services approach in two aspects First it does not assume that consumers preferences exhibit a consumption externality second it assumes complementarity in the sense that just as computers yield no utility without the supporting software the basic computer component does not yield utility without a complementary monitor component 1031 The basic model The components models were first introduced in Matutes and Regibeau 1988 and Economides 1989 The product Consider a product that can be decomposed into two perfect complements components For example a computer system can be decomposed into a basic unit and a monitor The basic unit and the monitor are perfect complements since a consumer cannot use one component without using the other Another example is a stereo system which is generally decomposed into an amplifier and speakers We denote the first component the basic unit by X and the second component the monitor by Y Firms and Compatibility There are two firms capable of producing both components which can be assembled into systems We denote by XA the first component produced by firm A and by YA the second component produced by firm A Similarly firm B produces components XB and YB With no loss of generality we simplify by assuming that production is costless Turning to compatibility we can readily see that since the components are perfect complements each consumer must purchase one unit of X with one unit of Y The question of compatibility here is whether a consumer can combine components from different manufacturers when he or she purchases and assembles the system Formally Page 270 Definition 104 1 The components are said to be incompatible if the components produced by different manufacturers cannot be assembled into systems That is systems XAYB and XBYA do not exist in the market 2 The components are said to be compatible if components produced by different manufacturers can be assembled into systems That is XAYB and XBYA are available in the market Consumers There are three consumers denoted by AA AB and BB with heterogeneous preferences toward systems We denote by and the price of component X and component Y produced by firm i respectively i AB Each consumer has an ideal combination of components That is if and then consumer AA would always choose system XAYA over XBYB consumer BB would choose system XBYB over XAYA and if the systems are compatible see Definition 104 then consumer AB would choose system XAYB A consumer who purchases system XiYj would pay a total price of for this system ij AB We denote by Uij the utility level of consumer ij whose ideal system is XiYj and assume that for λ 0 Thus in this simple model each consumer has a different ideal system under equal prices The utility function 1012 shows that a consumer purchasing his ideal system gains a net of prices utility level of 2λ If the system he buys has one component from his ideal system and one component from his less preferred system his net of prices utility level is reduced by λ Finally a consumer who purchases a system in which both components are produced by his less preferred manufacturer has a net of prices utility level of 0 Clearly given the threshold utility level of 0 no system will be purchased unless its total cost is lower than 2λ Page 271 1032 Incompatible systems Suppose that the components produced by different manufacturers are incompatible see Definition 104 so that only two systems are produced system XAYA and system XBYB We denote by qi the number of systems sold by firm i and by pi the price of system i both components i AB That is the price of system XAYA is and the price of system XBYB is Thus the profit function of firm i is π piqi i AB We look for a NashBertrand equilibrium in prices Formally Definition 105 An incompatiblecomponents equilibrium is a pair of price and a pair of quantities and such that for a given firm i chooses to maxpi st qi number of consumers maximizing 1012 by choosing system i ij AB Before characterizing the equilibria we can show that Lemma 101 There does not exist an equilibrium where one firm sells to all consumers Proof If firm A sells to all customers then it must set pA 0 But even at this price if for e 0 sufficiently small firm B sets pB e consumer BB would purchase system XBYB What Lemma 101 tells us is that if an equilibrium exists then it must be that one firm sells to two consumers while the other sells to one Therefore Proposition 1013 There exist three equilibria In one equilibrium firm A sells system XAYA to consumers AA and AB while firm B sells system XBYB to consumer BB In this equilibrium In the second equilibrium firm B sells system XBYB to consumers BB and AB while firm A sells system XAYA to consumer AA In the second equilibrium In the third equilibrium firm A sells system XAYA to consumer AA firm B sells system XBYB to consumer BB and consumer AB is not served In this equilibrium and In any equilibrium the firms profit levels are given by Proof Since the first two equilibria are symmetric it is sufficient to look at the first equilibrium We have to show that firm A cannot increase its profit by reducing its price to a level at which it would sell to all three consumers undercutting firm B That is Page 272 Similarly we have to show that firm B cannot increase its profit by reducing its price pB to pA where it would sell to two consumers BB and AB In fact one should also check a third possibility in which firm B deviates by reducing the price to a level where all the three consumers purchase system XBYB However such a deviation is not profitable since firm B has to set First note that our candidate equilibrium prices satisfy equations 1013 and 1014 so no firm would find it profitable to reduce its price Second no firm could profitably deviate by raising its price since if firm B raises its price above 2λ consumer BB will not purchase system XBYB Similarly if firm A raises its price above λ consumer AB will not purchase any system We still have to show that consumers AA AB and BA maximize their utility1012 by choosing system AA and that consumer BB maximizes utility by choosing system XBYB To do that we need to calculate the equilibrium utility levels of all customers Thus in equilibrium we have it that It is easy to verify that consumer AA would not purchase system BB since system BB would yield a utility level of Similarly consumer BB would not purchase system AA since system AA would yield a utility level of Also consumer AB would not purchase system BB since and both yield a net of prices utility level of λ Finally to show that constitute the third equilibrium note that if say firm A reduces its price to pA λ consumer AB buys system AA and we have the first equilibrium Since in all equilibria firm As profit level is a deviation will not occur We define the consumer surplus as the sum of consumers utilities Hence We define the economys welfare as the sum of firms profit levels and consumer surplus Thus The equilibrium socialwelfare level given in 1017 is simply the sum of the net of prices utility levels of all the consumers which equals Page 273 twice 2λ for consumers AA and BB who consume their ideal systems and λ for consumer AB who purchases the system XAYA but whose ideal Y component is YB 1033 Compatible systems When firms design their components to be compatible with components produced by the rival firm two more systems become available to consumers system XAYB and system XBYA We look for an equilibrium where each consumer buys assembles his ideal system In this equilibrium each firm i sells two units of component Xi and two units of component Yi i A B Definition 106 A compatible components equilibrium is the set of component prices and quantities of components sold by each firm such that for given and firm i chooses and to max st and are the number of consumers maximizing 1012 by choosing components Xi Yi respectively Proposition 1014 There exists an equilibrium where each consumer purchases his ideal system In this equilibrium all components are equally priced at and a firms profit levels are Proof Since firm A sells two components of X and one component of Y while firm B sells two components of Y and one component of X equilibrium prices should be at levels so that firms could not profitably reduce the price of one component in order to sell this component to additional customers For example in equilibrium firm A sells component XA to consumers AA and AB Reducing the price of YA to would induce consumer BB to buy component Y from firm A note that in order to attract consumers from the competing firms the price reduction should be at least λ However reducing a component price to zero cannot constitute a profitmaximizing deviation By symmetry firm B will not find it profitable to reduce its price to Finally since all prices are equal each consumer purchases his ideal brand yielding equilibrium utility levels of For this reason no firm would find it profitable to increase a components price since each consumer would not pay more than 2λ for a system Hence when all components are compatible the aggregate consumer surplus firms profit levels and the social welfare level are given Page 274 by Like equation 1017 equation 1019 demonstrates that the social welfare is the sum of the net of prices utility levels 1034 Compatibility versus incompatibility We now wish to examine the effects of components compatibility on firms profit and consumers utility levels aggregate consumers surplus and the social welfare Comparing 1015 with 1018 yields Proposition 1015 Consumers are never better off when the firms produce compatible components than when firms produce incompatible components However comparing Propositions 1013 with 1014 yields Proposition 1016 All firms make higher profits when they produce compatible components than when they produce incompatible components Also comparing 1017 with 1019 yields Proposition 1017 Social welfare is higher when firms produce compatible components In order to explain Proposition 1015 we need to compare the systems prices under the compatibility and incompatibility regimes given in Propositions 1013 and 1014 Under the incompatibility regime two consumers pay each λ for the system they buy Under compatibility each consumer pays 2λ for each system Hence total consumer expenditure under compatibility exceeds the expenditure under incompatibility by 2λ but the net of prices utility level of consumer AB the mixing consumer rises by only λ Thus firms extract a surplus that exceeds the aggregate utility gains from compatibility thereby reducing aggregate consumer surplus under the compatibility regime Proposition 1016 can be explained by the following First under compatibility the mixing consumer is willing to pay more because he can now buy his ideal system Second compatibility reduces price competition between the componentproducing firms since under incompatibility both firms are forced to lower the price of their system in order to attract the mixing consumer to choose their systems given that the systems are not ideal for this consumer This competition is relaxed when the components are compatible Page 275 Finally Proposition 1017 shows that the welfare gains derived from having firms increase their profits by making their components compatible exceeds the welfare loss to consumers from the high component prices under compatibility 1035 How firms design their components Proposition 1016 shows that firms collect higher profits when all components are compatible with the components produced by the rival firms than they collect when firms produce incompatible components We now ask whether an outcome where both firms choose to produce compatible components can be realized as an equilibrium for game in which firms choose both prices and the design of the components Consider a twostage game where in period I firms choose whether to design their components to be compatible with the components produced by the rival firm In period 2 given the design of the components firms compete in prices as described in subsections 1032 and 1033 The subgame perfect equilibrium for this game turns out to be very simple because the compatibility decision by one firm forces an externality on the rival firm in the sense that the compatibility of components is a symmetric relation meaning that if component XA is compatible with component YB then component YB is compatible with component XA In other words the market effect of having firm A make its XA component compatible with component YB is equivalent to having firm B make its YB component compatible with XA Similarly the outcome in which firm B makes its XB component compatible with firm As YA component is equivalent to firm As making its YA component compatible with Bs XB component It is important to note that this externality is a feature of the component approach discussed here but it does not occur in the supportingservices approach see Definition 103 That is in the supportingservices approach we can have it that machine A reads Bs software but not the other way around Thus given this externality feature of the components approach it is sufficient for one firm to decide on compatibility to produce a market outcome identical to that which would result from both firms deciding on compatibility Therefore Proposition 1016 implies that Proposition 1018 In the twostage game a subgame perfect equilibrium Fields compatible components Page 276 104 Exercises 1 Consider the supportingservices approach model developed in subsection 102 a For a given hardware price of brand A pA what is the price of computer B beyond which firm B would have a zero market share b Suppose that pA pB and suppose that the income of each consumer doubles to 2Y while hardware prices remain unchanged Calculate the effect this increase in incomes on i the market shares δA and δB and on ii the ratio of the number of software packages written for computer A to the number of software packages written for computer B 2 Consider the component approach analyzed in subsection 103 but assume that there are four consumers consumer AA consumer BB consumer AB and consumer BA a If the components are incompatible prove that no NashBertrand equilibrium in system prices pA and pB as defined in Definition 105 exists b If the components are compatible calculate the symmetric equilibrium prices of all components firms profit levels and consumers surplus 105 References Chou C and O Shy 1990 Network Effects without Network Externalities International Journal of Industrial Organization 8 259270 Chou C and O Shy 1993 Partial Compatibility and Supporting Services Economics Letters 41 193197 Chou C and O Shy 1996 Do Consumers Gain or Lose When More People Buy the Same Brand European Journal of Political Economy 12 309330 Church J and N Gandal 1992a Integration Complementary Products and Variety Journal of Economics and management Strategy 1 651676 Church J and N Gandal 1992b Network Effects Software Provision and Standardization Journal of Industrial Economics 40 85104 Church J and N Gandal 1993 Complementary Network Externalities and Technological Adoption International Journal of Industrial Organization 11 239260 Conner K and R Rumelt 1991 Software Piracy An Analysis of Protection Strategies Management Science 37 125139 David P 1985 Clio and the Economics of QWERTY American Economic Review 75 332336 Page 277 David P and S Greenstein 1990 The Economics of Compatibility Standards An Introduction to Recent Research Economics of Innovation and New Technology 1 342 Economides N 1989 Desirability of Compatibility in the Absence of Network Externalities American Economic Review 79 11651181 Farrell J and G Saloner 1986 Standardization and Variety Economics Letters 20 7174 Farrell J and G Saloner 1987 The Economics of Horses Penguins and Lemmings In Production Standardization and Competitive Strategies edited by L G Gable Amsterdam North Holland Farrell J and C Shapiro 1992 Standard Setting in HighDefinition Television Brookings Papers on Economic Activity Microeconomics 193 Gabel L 1991 Competitive Strategies for Product Standards London McGraw Hill Gandal N 1994 Hedonic Price Indexes for Spreadsheets and an Empirical Test of Network Externalities RAND Journal of Economics 25 160170 Katz M and C Shapiro 1985 Network externalities Competition and Compatibility American Economic Review 75 424440 Katz M and C Shapiro 1986 Technology Adoption in the Presence of Network Externalities Journal of Political Economy 94 822841 Kindleberger C 1983 Standards as Public Collective and Private Goods KYKLOS 36 377396 Matures C and P Regibeau 1988 Mix and Match Product Compatibility Without Network Externalities RAND Journal of Economics 19 221234 Rohlfs J 1974 A Theory of Interdependent Demand for a Communication Service Bell Journal of Economics 5 1637 PART IV MARKETING Page 281 Chapter 11 Advertising Hardly any business practice causes economists greater uneasiness than advertising L Telser Advertising and Competition Advertising is an integral part of our life Each one of us is constantly bombarded by advertising for products and services in a wide variety of forms We watch advertising on TV listen to advertising on the radio read ads in newspapers in magazines on outdoor billboards on buses and trains receive a large amount of socalled junk mail and we transmit advertising via wordofmouth and by wearing brandname labels on our clothes Despite this basic observation very little is understood about the effects of advertising Advertising is generally defined as a form of providing information about prices quality and location of goods and services Advertising differs from other forms of information transmissions like stock exchange data and guidebooks in two respects First the information is transmitted by the body who sells the product and second the buyer does not always have to pay to receive the information or pays a little with his or her value of time of watch a TV ad or to sort out the relevant ads in the Sunday newspaper What is the purpose of advertising We first need to acknowledge that advertising must serve a purpose for some agents in the economy sinceas a matter of factfirms governments and individuals spend large sums of money on advertising It is generally estimated that developed economies spend more than 2 percent of their GNPs on advertising see Schmalensee 1972 1986 The expenditure of firms on advertising is generally measured in terms of advertising expenditure divided by the value of sales These ratios vary drastically across products and industries The ratio of advertising expenditure to sales of vegetables may Page 282 be as low as 01 percent whereas for cosmetics or detergents this ratio may be as high as 20 to 60 percent There have been many attempts to correlate industry types product characteristics geographical locations and other characteristics with advertisingtosales ratios However in most cases advertising still remains a mystery since neither empirically nor theoretically can we explain why different firms spend different amounts on advertising For example Adams and Brock 1990 report that the Big Three car producers in the United States which are ranked among the largest advertisers in the country happen to have different advertisingtosales ratios In 1986 the largest producer GM which spent 285 million on advertising spent 63 per car whereas Ford spent 130 and Chrysler spent 113 per car though they spent less overall than GM This may hint of economies of scale in car advertising Earlier modern authors eg Kaldor 1950 held the idea that advertising is manipulative and reduces competition and therefore reduces welfare for two reasons First advertising would persuade consumers to believe wrongly that identical products are differentiated because the decision of which brand to purchase depends on consumers perception of what the brand is rather than on the actual physical characteristics of the product Therefore prices of heavily advertised products would rise far beyond their cost of production Second advertising serves as an entry deterring mechanism since any newly entering firm must extensively advertise in order to surpass the reputation of the existing firms Thus existing firms use advertising as an entrydeterrence strategy and can maintain their dominance while keeping abovenormal profit levels More recent authors Telser 1964 Nelson 1970 1974 and Demsetz 1979 proposed that advertising serves as a tool for transmitting information from producers to consumers about differentiated brands thereby reducing consumers cost of obtaining information about where to purchase their most preferred brand Nelson 1970 distinguishes between two types of goods search goods and experience goods Consumers can identify the quality and other characteristics of the product before the actual purchase of search goods Examples include tomatoes or shirts Consumers cannot learn the quality and other characteristics of experience goods before the actual purchase Examples include new models of cars and many electrical appliances with unknown durability and failure rates Note that this distinction is not really clearcut since we cannot fully judge the quality of a tomato until we eat it and we cannot fully judge the quality of a shirt until after the first wash What Nelson claims is that the effects of advertising may vary between these two groups of products because consumers do not depend Page 283 on information obtained from the manufacturers concerning search products since consumers find it by themselves However consumers do rely on advertisements when they purchase experience goods Several tests have also confirmed that advertising of experienced products is more intensive in terms of the ratio of advertising expenditure to sales than advertising of search goods The economics literature distinguishes between two types of advertising persuasive advertising and informative advertising Persuasive advertising intends to enhance consumer tastes for a certain product whereas informative advertising carries basic product information such as characteristics prices and where to buy it In the following two subsections we analyze these two types of advertising and ask whether from a social welfare point of view firms engage in too little or too much advertising 111 Persuasive Advertising In this subsection we analyze persuasive advertising That is advertising that boosts the industry demand for the advertised products We first investigate what the optimal advertising level is assuming that the demand for the good is monotonically increasing with the firms advertising level Then we ask whether from a social welfare point of view there is too much or too little advertising 1111 The monopolys profitmaximizing level of advertising Consider a monopoly firm selling a single product in a market where the demand curve is given by The parameter A denotes the firms expenditure on advertising Q and p denote the quantity demanded and the price for this product Thus the quantity demanded is monotonically increasing with the level of advertising A but at a decreasing rate since Denoting by ηAA p and ηpAp the demand advertising elasticity and price elasticity respectively and recalling subsection 321 where we showed the exponents of the variables in an exponential demand function illustrated in Figure 34 are the elasticities of the corresponding variables the reader can verify that Page 284 Let c denote the unit cost of the product The monopoly has two choice variables the price p and the advertising expenditure A Thus the monopoly solves The firstorder condition with respect to price is given by implying that The firstorder condition with respect to advertising level is given by implying that Equating equations 115 with 117 yields Equation 118 is known as the DorfmanSteiner 1954 condition Therefore Proposition 111 A monopolys profitmaximizing advertising and price levels should be set so that the ratio of advertising expenditure to revenue equals the absolute value of the ratio of the advertising elasticity to price elasticity Formally Thus a monopoly would increase its advertisingtosales ratio as the demand becomes more elastic with respect to the advertising is close to 1 or less elastic with respect to price is close to zero Page 285 1112 Too much or too little persuasive advertising Persuasive advertising was defined as a method of information transmission that boosts the demand for the advertised product Thus persuasive advertising makes the good attractive to consumers and therefore has the potential to increase welfare This does not imply that persuasive advertising must be truthful All that persuasive advertising does is to provide an image for the product that would induce the consumer to purchase the product in order to identify with the message or people portrayed in the ads Dixit and Norman 1978 have proposed an extremely simple method for evaluating the welfare effect of persuasive advertising Consider a simplified version of the demand function 111 where β 64 and For this case we assume that Taking the unit production cost to equal c 1 the monopolist chooses pM and AM to maximize The firstorder condition with respect to p is given by implying that pM 2 and hence Since the demand function has a constant elasticity the monopoly price is independent of the level of advertising The firstorder condition with respect to A is given by implying that AM 64 and hence In order to check whether the monopoly advertises at the socially optimal level we first need to calculate the consumer surplus associated with each advertising level The shaded area in Figure 111 shows the consumer surplus associated with a given advertising level A and the monopoly price pM 2 Hence for a given advertising level A the consumer surplus is given by Page 286 Figure 111 Consumer surplus for a given persuasiveadvertising level Assuming a monopoly price of pM 2 the firms profit level as a function of the level of advertising is given by The social planner takes the monopoly price pM 2 as given and chooses an advertising level A to The firstorder condition is given by Hence the socially optimal advertising level is A 242 64 AM Notice that this social optimum is not a firstbest optimum since a firstbest optimum requires marginal cost pricing Hence Proposition 112 Given a monopoly market structure the equilibrium level of persuasive advertising is below the socially optimal level Finally the model presented in this section is very special and is given for the purpose of introducing one method for evaluating the welfare effects of persuasive advertising We note here several problems concerning the robustness of Proposition 112 First is it appropriate to use the consumer surplus as a welfare measure when the demand utility is affected by the advertising level Second even if this measure is appropriate since the model is a partial equilibrium one the measure Page 287 does not capture the entire welfare effect associated with an increase in the demand for the advertised product That is an increase in the demand for one product would decrease the demand for other products say for substitute products Hence the change in consumers surplus in other markets should be taken into account 112 Informative Advertising Consumers often rely on information for their purchases Without advertising few consumers would be exposed to the variety of existing products the price distribution and the location of specific products As Nelson points out advertising can serve as a tool for transmitting this information to consumers and therefore should not be considered as an unnecessary activity In fact Benham 1972 has shown that prices are lower in markets where prices of eyeglasses are advertised than in markets where prices are not advertised The literature investigating the welfare effects of informative advertising concentrates on the conventional question of whether there is too little or too much informative advertising Butters 1977 develops a model in which firms advertise the price of a homogeneous product and finds that the aggregate advertising level determined in a monopolistic competition equilibrium is socially optimal Thus Butters shows that informative advertising need not always be detrimental Grossman and Shapiro 1984 consider a world of product differentiation where consumers who are located on the circumference of a circle see subsection 732 are able to recognize a brand only if the producer advertises This model provides ambiguous results about the excessiveness of informative advertising Thus the literature demonstrates that whether informative advertising is excessive or not depends on the specific functional form used for describing the industry Recently Meurer and Stahl 1994 developed a model in which some consumers are informed about two differentiated products and some are not and in which both advertising and prices are choice variables They show that social welfare may increase or decrease depending on the level of advertising We proceed by developing a very simple model to analyze this question Obviously the answer that will be given here is not robust However the purpose of developing this model is to present one approach for how to model this type of question Consider a singleconsumer singleproduct market Let p be the price of the product and assume that p is exogenously given eg p is regulated Let m denote the consumers benefit from purchasing one unit of the product Altogether we assume that the utility function of Page 288 the consumer is given by There are two rims producing the same product and offering it for sale at a price of p With no loss of generality assume that production is costless so that the only cost firms have to bear is the cost of sending an advertisement to the consumer Formally assume that each firm has a single decision variable which is whether or not to advertise The cost of advertising is given by a constant denoted by A The consumer may receive a total of 0 1 or 2 ads from the firms If the consumer receives one ad he buys the product from the firm that sent it If he receives no ads he buys none and if he receives two ads he splits the transaction equally between the firms that is he pays p2 to each firm Note that this assumption is similar to the assumption that the consumer flips a coin when he receives two ads thereby yielding an expected revenue of p2 to each firm Therefore the profit of firm i i 1 2 is given by The fact that a firm sends an ad does not imply that the consumer will indeed receive it For instance even if the firm invests A in a TV ad it is possible that the consumer will not be watching TV at the time that the ad runs on the air Formally let δ 0 δ 1 be the probability that a message sent by a certain firm would be received by the consumer Therefore the expected profit of firm i i 1 2 is given by Comparing the expected profits in the first and second rows in 1119 to the reservation profit of 0 yields Proposition 113 For a given value of p 1 at least one firm will engage in advertising if and only if Page 289 2 two firms will engage in advertising if Figure 112 illustrates the combinations of the receiving probability parameterδ and the ratio of price to advertising cost pA associated with having no firm one firm or two firms plying ads Clearly Figure 112 Equilibrium number of firms placing ads for a low receiving probability δ or for a high advertising cost relative to the price low pA no firm would place an ad As either δ or pA increase the number of firms placing ads also increases We now turn to the welfare analysis The problem solved by the social planner is to choose the number of firms that advertise in order to maximize the expected sum of consumer surplus and firms profits First observe that if both firms advertise the probability that at least one firm would sell is 2δ1 δ δ2 δ2 δ which is twice the probability that one ad will be received while the other will not plus the probability that both ads are received Fondly the expected social welfare as a function of the number of ads is given by If we observe that p does not appear in 1120 it easy to infer that as long as p m a market failure is likely to occur This Page 290 happens because firms do not capture the entire consumer surplus and therefore will underadvertise compared with what a social planner would choose Therefore in order to check whether too many firms engage in advertising from a social viewpoint we set p m implying that all consumer surplus is absorbed in the firms profits In this case 1120 implies that it is socially optimal to have two firms sending ads rather than a single firm if and only if mA pA 1δ1 δ However Proposition 113 implies that a weaker parameter restriction is needed for having an equilibrium where two firms send ads That is mA pA 2δ2 δ Hence in Figure 112 the area between the curves given by 2δ2δ pA 1δ1 δ represents the parameter range where both firms advertise in equilibrium but it is socially optimal to have only one firm engaged in advertising Proposition 114 In a model where some placed ads do not reach the consumer there exists a parameter range 2δ 2δ pA 1δ1δ where too many firms engage in advertising from a social welfare point of view Finally what happens when the advertising technology improves in the sense that there is a higher probability ads sent to consumers arrive Figure 112 shows that when the upper curve shifts upward with no bounds implying that for high values of δ it is never socially optimal to have two firms engage in advertising The intuition is as follows Since sending ads is costly and since implies that ads are always received then one firm advertising is sufficient to have the consumer receive the information about the product 113 Targeted Advertising The literature on advertising assumes that advertising is either persuasive or informative That is the nature of advertising is always treated as exogenously given thereby ignoring the question of how firms choose the content for their advertising The underlying observation is that societies are composed of heterogeneous consumers with different rankings preferences over products Thus firms are unable to advertise and sell their brands to all types of consumers and therefore must limit the scope Of their advertising by choosing a narrow group of consumers to which their advertising appeals There may be three reasons for that First it is impossible to classify products attributes that are highly valued by all consumers Second given the high cost of advertising firms and advertising agencies may find it profitable to narrow the scope of advertising to a limited group Page 291 of consumers Third ignoring advertising costs since product differentiation may facilitate price competition firms may intentionally choose to target a limited consumer group The purpose of this section is to propose a framework for modeling firms choice of advertising methods and the resulting targeted consumer group where firms advertising must be confined to choosing a single advertising method and therefore a single consumer target group For example a firm may choose to advertise its brand by emphasizing one attribute of the product that is preferred by at least one consumer group but is not found in a competing brand Alternatively instead of advertising the products attributes a firm may target its advertising to a certain age group young or old or to inexperienced consumers and ignore the attributes quality differences among the competing brands 1131 Firms and consumers There are two firms denoted by i i 12 producing differentiated brands which we will refer to as brand 1 and brand 2 respectively There are two types of buyers There are N consumers who are firsttime buyers that we call the inexperienced consumers In addition there are E consumers who have purchased the product before and whom we call experienced consumers Figure 113 illustrates how the consumer population is divided between consumer types We assume that the N inexperienced consumers N inexperienced consumers θE experienced brand 1 oriented θE experienced brand 1oriented 1 θE experienced 1 θE experienced brand 2oriented brand 2oriented Figure 113 Targeted advertising Experienced versus inexperienced consumers out of total population Left E N Right E N group of experienced consumers is divided into two subgroups those who prefer to purchase brand 1 over brand 2 and those who prefer brand 2 over brand 1 Let θ 0 θ 1 be the fraction of brand 1oriented consumers among experienced consumers Therefore 1 θ is the Page 292 fraction of brand 2oriented consumers among experienced consumers Thus out of a total of E experienced consumers there are θE brand 1oriented and 1 θE brand 2oriented consumers 1132 Advertising methods There are two advertising methods A firm can use persuasive advertising a strategy denoted by P Alternatively a firm can use informative advertising a strategy denoted by I Thus each firm i chooses si from an action set given by For our purposes we assume that no firm can employ more than one advertising method that is a firm can choose P or I but not both One justification for such a strong assumption would be that advertising agencies tend to specialize in a single advertising method or philosophy Therefore if a firm would like to use both advertising methods it has to employ two advertising agencies which may increase cost more than profit To simplify our model we assume that choosing advertising methods is the only strategic variable available to firms Thus in this model we ignore prices and assume that firms seek to maximize the number of consumers buying their brand We denote by the vector of profit levels which equals the number of customers buying from each firm We make the following assumption Assumption 111 1 Persuasive advertising attracts only inexperienced consumers Formally if firm i chooses si P then a if firm j does not use persuasive advertising then all inexperienced consumers purchase brand i that is πi N if b if both firms use persuasive advertising then all inexperienced consumers are equally divided between the two firms that is πi N2 if sj p 2 In formative advertising attracts only the experienced consumers who are oriented toward the advertised brand Formally if firm 1 chooses s1 I then π1 θE and if firm 2 chooses s2 I π2 1 θE Table 11 1 demonstrates the profit level of each firm and the industry aggregate profit under all four possible outcomes s1 s2 We look for a Nash equilibrium see Definition 24 in the above strategies Page 293 Profit Outcome P P P I I P I I π1d N2 N θE θE π2 N2 1θE N 1θE π1 π2 N N 1 θE θE N E Table 111 Profits for firms under different advertising methods Proposition 115 1 A necessary condition for having both firms using persuasive ad vertising is that the number of inexperienced consumers exceeds the number of experienced consumers N E In this case P P is a unique equilibrium if 2 A necessary condition for having both firms using informative ad vertising is that the number of experienced consumers is more than twice the number of inexperienced consumers E 2N In this case I I is a unique equilibrium if 3 If brand I is unpopular among experienced users then firm I uses persuasive advertising and firm 2 uses informative advertising Formally PI is an equilibrium if 4 If brand I is sufficiently popular among experienced users then firm I uses informative advertising and firm 2 uses persuasive ad vertising Formally IP is an equilibrium if Proposition 115 is illustrated in Figure 114 The upper part of Figure 114 corresponds to part I of Proposition 115 where the number of experienced consumers is lower than the number of inexperienced con sumers Both firms use persuasive advertising when the brands have similar popularity among experienced users As the number of experienced consumers gets below E N2 the entire θ range corresponds to P P where both firms use persuasive advertising That is for every popularity parameter θ the unique equilibrium is P P The lower part of Figure 114 corresponds to part 2 of Proposition 115 where the number of experienced consumers is more than twice the number of inexperienced consumers In this case both firms use informative advertising unless one brand is very popular among the experienced consumers compared with the other brand Then a firm would use persuasive advertising only if its brand is very unpopular among the experienced consumers Finally as the number of experienced Page 294 Figure 114 Informative versus persuasive advertising consumers increases with no bounds the entire popularity parameter θ range corresponds to having both firms using only informative advertising Proof of Proposition 115 Part 1 We look at firm 1 In this equilibrium π1P P N2 If firm 1 deviates and chooses s1 I then π1 I P θE Therefore a deviation is not profitable for firm I if N2 θE or if θ N2E Similarly firm 2 will not deviate if 1 θ N2E or θ 1N2E In order for this region to be nonempty we must have it that 1 N2E N2E implying that E N Part 2 In this equilibriumπ1I I θE If firm I deviates and chooses s1 P then π1 I P N Hence firm I will not deviate if θE N or if θ NE Similarly firm 2 will not deviate if 1 θ NE or θ 1 NE In order for this region to be nonempty we must have it that NE 1 NE implying that E 2N Part 3 For firm 1 π1P I N If firm I deviates to s1 I then π1 I I θE Hence firm 1 will not deviate if N θE or if θ NE For firm 2 π2PI 1 θE If firm 2 deviates to s2 P then π2P P N2 Hence firm 2 will not deviate if 1 θE N2 or θ 1 N2E Altogether θ min NE 1 N2E Finally Part 4 can be proved in a similar way and we leave it as an exercise to the reader 114 Comparison Advertising Comparison advertising is defined as one in which the advertised brand and its characteristics are compared with those of the competing brands Page 295 1141 Comparison advertising an overview In the United States no law ever prevented the use of comparison advertising However advertisers were reluctant to use it Boddewyn and Marton 1978 Only in the early 1970s did television networks begin to extensively broadcast comparison advertisements Since then comparison ads have become popular in the printed media as well as in the broadcast media The EEC also began to address the issue of comparison advertising in the late 1970s suggesting that comparison advertising should be legal as long as it compares material and verifiable details and is neither misleading nor unfair The principle advantage of comparison advertising is that the information contained in a comparison advertisement provides consumers with lowcost means of evaluating available products Barnes and Blakeney 1982 In addition comparison advertising makes the consumers more conscious of their responsibility to compare before buying It also forces the manufacturer to build into the products attributes consumers want and eventually to produce a better product There are arguments suggesting that comparison advertising does not assist consumer comparisons because the comparison will lack objectivity since the advertiser will select only those aspects of his brand that are superior to those of the competitors The critics consider that the risk of consumer confusion and deception is great in comparison advertising partly because of information overload In most countries where comparative advertising is legal it is closely monitored and regulated by government agencies Different studies suggest different figures on the relative use of comparative advertising Muehling Stoltman and Grossbart note that around 40 percent of all advertising is comparative Others Pechmann and Stewart 1990 and references suggest that the majority of all ads are indirectly comparative 60 percent as opposed to 20 percent that contain direct comparative claims the rest are noncomparative 1142 Strategic use of comparison advertising The model developed in section 113 can be modified to capture the effects of comparative advertising Assume that each firm has an action set given by where C means that a firm uses comparison advertising and A means that the firm advertises its product without comparing it to the competing brand Page 296 Following Assumption 111 we assume that Assumption 112 1 Plain noncomparative advertising A attracts only the inexperienced consumers 2 Comparison advertising C attracts only the experienced consumers who are oriented toward the advertised brand Thus plain noncomparative advertising is intended to inform consumers about the existence of the product by informing the consumer about a specific brand The drawback of plain advertising is that it also attracts new consumers of the wrong type In contrast comparison advertising informs the experienced misplaced consumers wrongbrand users about the difference between the brand they have purchased in the past and their ideal brand Thus a firm uses the comparisonadvertising strategy to attract experienced users who are oriented toward its brand The intuition behind Assumption 112 is simple It is likely that a comparison advertisement is meaningless for the inexperienced consumer simply because a nonuser may not understand the way the product and its features operate Thus an inexperienced consumer will not comprehend an ad involving a comparison of the brands attributes Assumption 112 suggests that the relevance of comparison advertising is a consequence of prior experience with the product itself Assumption 112 also suggests that plain advertising is not very relevant irrelevant in our extreme case to the experienced user since an experienced user definitely knows about the existence of the product and its basic features Although Assumption 112 sounds very intuitive it has not been tested In fact many experiments cited in the references eg chapter 7 of Boddewyn and Matron 1978 tend to find very little difference in the effects produced by comparative and by noncomparative advertising However none of these tests attempted to test them on experienced and firsttime buyers separately Applying Proposition 115 to the present case yields Proposition 116 1 Comparison advertising is used by both firms when the majority of the potential consumers are experienced That is when E 2N 2 Comparison advertising will not be used if the number of inexperienced consumers is larger than the number of inexperienced consumers That is when E N Page 297 3 Comparison advertising is used by the popular firm producing the more popular brand among the experienced consumers That is a firm would use comparison advertising when the fraction of experienced consumers oriented toward its brand is large 115 Other Issues Concerning Advertising 1151 Advertising and quality Information about prices of products is often easier to acquire than information about the quality of products It is relatively easy although costly to find out the distribution of prices for TV sets However it is difficult to find out the frequency of repair of various TV brands for the simple reason that producers do not release these data to consumers Several authors questioned whether information on quality can be transmitted via advertising That is can advertising correctly inform consumers on the quality of the product If the answer is yes then one should ask what the exact relationship is between advertising and the quality of the advertised product Advertising a search good if it occurs is likely to be honest because lies will be detected immediately Thus false advertising of search goods may hurt firms reputations rather than enhance them This need not be the case for experience goods for which producers may gain from false advertising at least in the short run Producers of experience goods will attempt to develop all kinds of persuasive methods to get consumers to try their products There are few analytical models attempting to find the link between advertising and quality Schmalensee 1978 finds that lowquality brands are more frequently purchased and that firms producing lowquality products advertise more intensively Thus there is a negative correlation between the intensity of advertising and the quality of the advertised product Kihlstrom and Riordan 1984 develop a twoperiod model in which high and lowquality products are sold and high quality firms have an incentive to advertise in order to trap the consumers seeking to purchase highquality products in the second period ie trap repeat buyers Their model finds a positive correlation between advertising intensity and the quality of the advertised product On this line which is similar to the signaling model of subsection 846 Milgrom and Roberts 1986 develop a signaling model in which a high level of advertising is used as a signal sent by highqualityproducing firms to those consumers who desire to purchase highquality products Bagwell 1994 and Bagwell and Ramey 1994 argue that efficient firms operating under increasing Page 298 returns tend to spend large amount on advertising to convince buyers that large sales will end up with lower prices due to lower cost Thus efficient firms would spend more on advertising than less efficient firms to reveal their cost identity to the buyers 1152 Advertising and concentration Basic intuition may lead us to think that in a near competitive industry with a large number of firms no firm would have an incentive to advertise since persuasive advertising may boost the demand facing the industry but may have only a small effect on the demand facing the advertising firm Thus a free rider effect will generate little advertising Recognizing this effect leads advertising associations in some countries to advertise how good advertising can be This kind of argument generates the testable hypothesis that intensive advertising high advertising expendituretosales ratio is associated with the more concentrated industries concentration measures are analyzed in section 81 Orenstein 1976 summarized early empirical tests that attempted to investigate a connection between advertising and concentration From a theoretical point of view this hypothesis can be explained by an increasingreturns type of argument Kaldor claimed that if one takes an industry in which advertising is prohibited and then allows advertising the larger firms would increase their advertising expenditure at a faster rate than the smaller firms thereby increasing industry concentration However Telser 1964 demonstrated very little empirical support for an inverse relationship between advertising and competition In addition Orenstein 1976 tested for increasing returns in advertising say resulting from a falling advertising cost associated with an increase in advertising volume but showed very little evidence in favor of this hypothesis For a very comprehensive recent empirical and theoretical study of the association between industry structure concentration and advertising intensity the reader is referred to Sutton 1991 Several authors including Sutton 1974 suggested that the relationship between advertising and concentration need not be always monotonically increasing and that there can exist a certain concentration level at which advertising is most intensive That is the relation between advertising and concentration may take the form of an upsidedown Ushaped function Sutton suggested that industries with low concentration are associated with low incentives to advertise together with low opportunity by incentive Sutton meant the extra profit generated by extra advertising whereas by opportunity he meant the success of the advertising However Sutton suggested that in highly concen Page 299 trated industries both the incentives and the opportunity are lower than in mediumlevel concentrated industries because profit expectations tend to be higher in mediumconcentration industries 1153 Advertising and prices Despite the fact that there is no significant evidence for the association between concentration and advertising intensity there is however some evidence on how advertising affects prices Benham 1972 found that the average price of eyeglasses in states where advertising eyeglasses is prohibited is around twice the average price of eyeglasses in states where eyeglasses are advertised A similar test regarding the introduction of toy advertising on television suggests a sharp price reduction following this introduction How can we explain this observation that high advertising intensity is associated with lower price but not necessarily in a reduced market concentration We demonstrate it by the following simple example Let us first assume that there is only one firm monopoly selling a particular good whose period 0 demand is given by Q a0 p where a0 is or is positively related to the period 0 level of advertising by the monopoly Let A denote the advertising cost We assume a simple form of increasingreturns technology represented by the following cost function Thus for a given advertising level A the variable cost is discontinuous at the output level Q Figure 115 illustrates that the marginal production cost falls to cL at output levels exceeding Q reflecting a situation where at high output levels the firm uses a different production method say employing assembly lines to assemble products or shipping production overseas to lowwage countries We saw in section 51 that the period 0 monopoly equilibrium is at a production level of QM0 a0 c0 and a price level of pM0 a0 cH2 Now suppose that in period 1 the monopoly intensifies its advertising effort and spends A1 A0 on advertising We assume that a higher level of advertising shifts the demand to Q a1 p where a1 a0 Figure 115 shows that the new equilibrium is associated with an output level QM1 a1 cL2 and a price of pM1 a1 cL2 Comparing the prices associated with the two advertising levels yields Proposition 117 Monopoly price pM0 pM0 if and only if cH cL a1 a0 That is advertising reduces the monopoly price if and only if the reduction in marginal cost associated with a higher production level exceeds the level of change in the demand Page 300 Figure 115 Advertisinginduced demand increase and falling prices We have ignored the question of whether advertising is profitable for this monopoly since it simply depends on how the period 1 advertising expenditure relates to the period 0 advertising level ie on the magnitude of A1 A0 If this difference is relatively low then the monopoly will advertise and price will fall if the condition in Proposition 117 is fulfilled If the difference in advertising expenditure is large then the monopoly may choose not to increase its advertising level In any case we have shown that it is possible to have a situation where prices fall or rise when advertising increases but the industry concentration level remains unchanged in this case concentration remains at the level of 100 percent The conclusion from this experiment demonstrates a very well known econometric problem in which looking at data on prices and quantities cannot reveal what has happened to concentration since prices and quantities may be affected by demand and production cost changes at the same time 116 Appendix Advertising Regulations Advertising regulation has two purposes 1 Regulation prevents firms from using advertising in a way that limits the competition among the firms in the industry 2 Regulation is intended to protect consumers from false advertising and misrepresentations In addition some negative advertising Page 301 such as the labels on clothes or smokingalert labels on cigarettes is sometimes mandated by governments The main difficulty in establishing advertising regulations stems from the fact that these two goals may in some cases conflict with one another That is in order to protect the consumer against misrepresentations the FTC or the local government have to limit the scope of advertising However restricting advertising may hamper the operation of the competitive process A second difficulty in regulating advertising stems from the fact that many countries allow free speech including commercial free speech implying that producers are free to advertise their products and services Yet producers of product or services tend to misrepresent their products and services thereby leading some consumers to believe that they buy what they want although they actually do not In the following subsections we discuss some advertising regulations in two large markets The United States and the EC The interested reader is referred to Barnes and Blakeney 1982 McManis 1988 and Maxeiner and Schotthöfer 1992 for extensive discussions and analysis of countryspecific advertising regulations 1161 The United States We focus most of our discussion on the United States since advertising is used most intensively in the United States and paradoxically advertising is heavily regulated in the United States In the United States federal state and local governments independently regulate advertising Concurrent regulation is not contradictory since state laws should not conflict with federal laws In practice advertising laws differ from state to state Federal advertising legislation is found in two major laws the Federal Trade Commission Act and the Trademark Lanham Act In practice the FTC issues advertising guidelines to the industry States create their own versions of the FTC Act Finally the private sector is also active as a self regulator by imposing many rules via organizations such as the Consumers Union and Better Business Bureaus Under the First Amendment to the US Constitution freedom of speech is protected However freedom of speech applies only to truthful advertising that is false advertising is not protected The Trademark Lanham Act prohibits the use of false designations of origin and false or misleading descriptions of fact and the representation of a fact This includes the prohibition of the creation of confusion about the origin sponsorship and approval of goods and services The FTC Act prohibits any unfair methods of competition including dis Page 302 semination of false advertisements There is the question of which ads constitute false or misleading advertising First misleading advertising has to be material ie it should affect the consumers decisions Second the claims or implied claims made in the ad have to be false where omissions do not constitute false advertising Third the ads have to mislead a substantial fraction of the audience where the audience is expected to have a reasonable interpretation The FTC requires that the advertisers advertising agencies will have bases for their advertised objective claims Subjective claims such as this product has changed my life for the good or for the bad need not be substantiated This guide is particularly important for the case of comparison advertising which is perfectly legal even somewhat encouraged in the United States but all claims must be substantiated Finally in the United States there are special federal laws that address special products and services For example advertising cigarettes on TV is prohibited Special regulations prevail for advertising financial investments and drugs 1162 The European Community Advertising in Europe is generally regulated by national laws Regulation by the EC takes the form of directives to governments meaning that the member countries would have to adopt their own laws in order to achieve the directed results The EC Treaty guarantees the freedom of movement of goods and services across member states This implies the freedom of transnational advertising Thus the idea is to promote a market favorable to all member states products The EC directive toward TV and radio advertising is intended to limit the ads separable from the programs to a maximum of 20 percent of the broadcasting time The ads should not be discriminative on the basis of nationality or any other basis Cigarette advertising is prohibited and advertising alcoholic beverages on TV is restricted In addition comparison advertising is legal as long as it is based on substantiated grounds Australia also allows comparison advertising based on testable claims see Barnes and Blakeney 1982 Finally the EC has also issued some directives concerning misleading advertising thereby encouraging member states to adopt measures in order to prevent it 117 Exercises 1 Congratulations You have been appointed to become a CEO of UGLY Inc the sole producer of facial oil skinlife extender Your first as Page 303 signment is to determine the advertising budget for next year The marketing department provides you with three important information items a The company is expected to sell 10 million worth of the product b It is estimated that a 1 percent increase in the advertising budget would increase quantity sold by 005 percent c It is also estimated that a 1 percent increase in the products price would reduce quantity sold by 02 percent a How much money would you allocate for advertising next year b Now suppose that the marketing department has revised its estimation regarding the demand price elasticity to 1 percent increase in price resulting in a reduction in quantity sold by 05 percent How much money would you allocate to advertising after getting the revised estimate c Conclude how a change in the demand price elasticity affects advertising expenditure 2 In Future City there are two fortunetellers Ms α and Mr β Each fortuneteller charges a fixed regulated fee of 10 for one visit Let Ai denote the advertising expenditure of fortuneteller i i α β The number of clients visiting each teller per unit of time is denoted by ni i α β We assume that ni depends only on the advertising expenditure of both tellers Formally let Thus the number of clients visiting teller α increases with αs advertising expenditure and decreases with βs advertising expenditure Altogether assume that each fortuneteller i has only one choice variable which is the advertising level and therefore chooses Ai to maximize the profit given by a Compare the number of visitors and the profit level of each fortuneteller when Aα Aβ 1 and for Aα Aβ 2 What can you conclude about the role of advertising in this city b Calculate and draw the bestresponse function of teller β as a function of the advertising expenditure of teller α In case you forgot how to define bestresponse functions we first used them in section 61 c Calculate the tellers advertising level in a Nash equilibrium d In view of your answer to a is the Nash equilibrium you found in c optimal for the fortuneteller industry e Is the equilibrium you found stable 3 Prove part 4 of Proposition 115 Hint Follow the same steps as in the proof of part 3 Page 304 118 References Adams W and J Brock 1990 The Automobile Industry In Structure of American Industry edited by W Adams New York Macmillan Publishing Company Bagwell K 1994 Advertising and Coordination Review of Economic Studio 61 153172 Bagwell K and G Ramey 1994 Coordination Economics Advertising and Search Behavior in Retail Markets American Economic Review 84 498517 Barnes S and M Blakeney 1982 Advertising Regulation Sydney The Law Book Company Benham L 1972 The Effects of Advertising on the Price of EyeGlasses Journal of Law and Economics 15 337352 Boddewyn J J and K Marton 1978 Comparison Advertising New York Hastings House Publishers Butters G 1977 Equilibrium Distributions of Sales and Advertising Prices Review of Economic Studies 44 465491 Demsetz H 1979 Accounting for Advertising as a Barrier to Entry Journal of Business 52 345 360 Dixit A and V Norman 1978 Advertising and Welfare The Bell Journal of Economics 9 117 Dorfman R and P Steiner 1954 Optimal Advertising and Optimal Quality American Economic Review 44 826836 Grossman G and C Shapiro 1984 Informative Advertising With Differentiated Products Review of Economic Studies 51 6381 Kaldor N 1950 The Economic Aspects of Advertising Review of Economic Studies 18 127 Kihlstrom R and M Riordan 1984 Advertising as a Signal Journal of Political Economy 92 427450 Maxeiner J and P Schotthöfer 1992 Advertising Law in Europe and North America Deventer The Netherlands Kluwer Law and Taxation Publishers McManis C 1988 Unfair Trade Practices in a Nutshell St Paul Minn West Publishing Co Meurer M and D Stahl 1994 Informative Advertising and Product Match International Journal of Industrial Organization 12 19 Milgrom P and J Roberts 1986 Price and Advertising Signals of Product Quality Journal of Political Economy 94 796821 Muehling D J Stoltman and S Grossbart 1990 The Impact of Comparative Advertising on Levels of Message Involvement Journal of Advertising 19 4150 Page 305 Nelson P 1970 Information and Consumer Behavior Journal of Political Economy 78 311329 Nelson P 1974 Advertising as Information Journal of Political Economy 82 729754 Orenstein S 1976 The Advertising Concentration Controversy Southern Economic Journal 43 892902 Pechmann C and D Stewart 1990 The Effects of Comparative Advertising on Attention Memory and Purchase Intentions Journal of Consumer Research 17 180191 Schmalensee R 1972 The Economics of Advertising Amsterdam NorthHolland Schmalensee R 1978 A Model of Advertising and Product Quality Journal of Political Economy 86 485503 Schmalensee R 1986 Advertising and Market Structure In New Developments in the Analysis of Market Structure edited by J Stiglitz and G Frank Matthewson Cambridge Mass MIT Press Shapiro C 1980 Advertising and Welfare Comment Bell Journal of Economics 11 749752 Sutton J 1974 Advertising Concentration Competition Economic Journal 5669 Sutton J 1991 Sunk Costs and Market Structure Cambridge Mass MIT Press Telser L 1964 Advertising and Competition Journal of Political Economy 72537562 Page 307 Chapter 12 Quality Durability and Warranties Anybody can cut prices but it takes brains to make a better article Philip D Armour 18321901 We observe that products within the same category are distinguished by a wide variety of characteristics Cars for example are differentiated by engine size horse power gas consumption body size number of doors body shape sedan vs hatchback transmission manual vs automatic and luxurious components such as air conditioning radio seat covers electric windows electric seats We tackled the issue of product differentiation in chapter 7 where we analyzed markets with firms target brands for different consumer populations and showed that product differentiation facilitates price competition In this chapter we wish to focus on one aspect of product differentiation that we call quality The only aspect of quality not explicitly analyzed is the risk health hazard involved in using the product see Oi 1973 We also confine part of the analysis in this chapter to one particular aspect of quality that we call durability The reason for focusing on durability separately from quality is that durability is related to the time dimension which has a direct impact on the frequency of repeated purchase by consumers For this reason some economists have argued that market structure has a strong effect on the durability aspect of the product but not necessarily on other quality aspects of the product In general it is hard to point out what constitutes the quality of a certain product since quality has many dimensions Using the exam Page 308 ple of the car we note that quality could mean acceleration frequency of maintenance frequency of repair comfort and safety Any reader of the consumer magazines will notice that consumer magazines rarely recommend one brand over all others for the simple reason that quality has many dimensions That is recommendations for choosing a certain brand are generally given conditionally on the specific needs of the user In most cases consumer magazines provide the readers with tables for comparing from ten to thirty features among the popular brands Hence in general brands are noncomparable on the basis of quality since each brand can be highly ranked because it has some features that are not available with other brands For this reason since multidimensional modeling of quality is very difficult we will follow the literature and assume that quality can be measured by a real number Thus we assume that a higher quality product is indexed by a higher real number Using this simplified measure of quality we analyze in section 121 Personal Income and Quality Purchase the relationship between consumer income distribution and the quality of products they purchase Section 122 Quality as Vertical Product Differentiation explains why firms produce brands with different qualities Section 123 Market Structure Quality and Durability discusses a thirtyyearold still ongoing debate about whether monopoly firms produce a less durable product than firms under competition Section 124 The InnovationDurability Tradeoff analyzes the effect of product durability on the frequency of introduction of new improved products Section 125 The Market for Lemons analyzes the market for used cars and demonstrates how the existence of bad cars can drive good cars from the usedcar market Section 126 QualitySignaling Games demonstrates how highquality firms can set their price structure in order to signal the quality of their products Section 127 Warranties analyzes the role that warranties can play when the quality of the product is unknown prior to the actual purchase In the appendix section 128 provides a short summary of productsliability laws 121 Personal Income and Quality Purchase We provide now a short illustration of how the level of personal income affects the quality of brands purchased by differentincome consumers In a series of models Gabszewicz and Thisse 1979 1980 and Shaked and Sutton 1982 use the following model to determine what the levels of qualities are and the number of different quality brands that are produced in an industry with free entry and exit For the sake of brevity we skip the analysis of the firms and concentrate only on consumers Page 309 Consider an industry with two firms producing brands with different qualities quality level k H and quality level k L H L 0 There are two consumers denoted by i i 12 The income of consumer 1 is given by I1 and the income of consumer 2 by I2 where I1 I2 0 Thus consumer 1 is the highincome consumer and consumer 2 is the lowincome consumer Each consumer buys only one unit of the product The utility level of consumer i i 1 2 is given by This utility function has the property that for given prices the marginal utility of quality rises with an increase in the consumers income The following proposition demonstrates how differentincome consumers are assigned to different quality products under the utility function given in 121 Proposition 121 1 If the lowincome consumer buys the highquality brand then the highincome Consumer definitely buys the highquality brand 2 If the highincome consumer buys the lowquality brand then the lowincome consumer definitely buys the lowquality brand Proof To prove part 1 let Uik denote the utility level of consumer i when he buys the brand with quality k We want to show that From 121 we have it that since consumer 2 buys the highquality brand then it must be that Hence Since I1 I2 we have it that Therefore This concludes the proof for the first part The second part is left as an exercise in section 129 Page 310 There have been several applications for the model presented above Gabszewicz and Thisse 1979 1980 and Shaked and Sutton 1982 present models based on the utility function 121 with more than two possible quality levels and show that even under free sequential entry only a small number of differentquality brands will be produced 122 Quality as Vertical Product Differentiation In subsection 731 we introduced the Hotelling location address approach to product differentiation We interpreted the location of each consumer as his preference for say a certain degree of sweetness desired in a chocolate bar where distance between a consumer and the firm is proportional to the consumers disutility from the specific brand it sells Another interpretation for the Hotelling model is simply the physical location of two stores where consumers must bear per unitofdistance transportation cost In this section we modify the Hotelling model to capture quality differences among differentiated brands 1221 Vertical differentiation in the basic Hotelling model The Hotelling model developed in subsection 731 was classified as a model of horizontal differentiation for the simple reason that given that the firms are located in the same street as the consumers there always exist consumers who would rank the two brands differently That is in the Hotelling model assuming that all brands are equally priced the consumer who is closer to firm A than to firm B would purchase brand A whereas a consumer who is closer to firm B would purchase brand B Thus given equal prices brands are not uniformly ranked among all consumers and for this reason we say that the brands are horizontally differentiated Phlips and Thisse 1982 emphasized the distinction between horizontal and vertical product differentiation in the following way Definition 121 1 Differentiation is said to be horizontal if when the level of the products characteristic is augmented in the products space there exists a consumer whose utility rises and there exists another consumer whose utility falls 2 Differentiation is said to be vertical if all consumers benefit when the level of the products characteristic is augmented in the product space Page 311 Figure 121 illustrates a simple diagrammatic comparison between horizontal and verticalquality differentiation for a comprehensive discussion of horizontal and vertical differentiation see Beath and Katsoulacos 1991 In Figure 121 all consumers are located between points 0 and 1 Figure 121 Horizontal versus vertical differentiation Up horizontal differentiation Down vertical differentiation The upper part of Figure 121 is the same as the Hotelling horizontaldifferentiation model displayed in Figure 77 In this case given equal prices the consumers located near firm A prefer brand A over brand B whereas consumers located near brand B prefer brand B over brand A In contrast the lower part of Figure 121 illustrates an industry with vertically differentiated brands where all consumers prefer brand A over brand B since all consumers are located closer to A than to B 1222 A modified Hotelling verticaldifferentiation model The basic Hotelling model developed in subsection 731 is based on preferences given in 717 and refers to the street illustrated in Figure 121 In what follows we modify the utility function 717 so that instead of having consumers gain a higher utility from the nearby brand all consumers would have their ideal brand located at point 1 on the 01 interval This modification would allow us to model product differentiation where firms still locate on the 01 interval and not outside this interval There is a continuum of consumers uniformly distributed on the interval 01 There are two firms denoted by A and B and located at points a and b from the origin respectively Figure 122 illustrates the location of the firms on the 01 interval The utility of a consumer located at point x and buying brand i i A B is defined by where pA and pB are the price charged by firm A and B respectively Page 312 Figure 122 Vertical differentiation in a modified Hotelling model We seek to define a twoperiod game where firms choose location in the first period and choose price in the second period after locations have been fixed Before defining the game let us solve for a NashBertrand equilibrium in prices assuming fixed locations as illustrated in Figure 122 Let denote a consumer who is indifferent to whether he or she buys from firm A or firm B Assuming that such a consumer exists and that consumer locates between the two firms that is the location of the indifferent consumer is determined by Thus the utility of consumer indexed by from buying brand A equals his utility from buying brand B Therefore assuming that the number of consumers buying from firm A is whereas the number of consumers buying from firm B is Solving for from 123 yields Figure 123 provides a graphic illustration of how is determined The lefthand side of Figure 123 illustrates the utility for a consumer located at any point when he or she buys brand A and when he or she buys brand B assuming that pB pA By definition for the consumer located at the utility from buying A equals the utility from buying B Moreover Figure 123 shows that all consumers located on gain a higher utility from purchasing brand A lower quality than from purchasing brand B Similarly all consumers located on gain a higher utility from purchasing brand B higherquality brand than from purchasing brand A Note that as in subsection 731 we assume here that consumers always buy one unit from firm A or from firm B In contrast assuming a reservation utility of zero would generate a group of consumers who do not purchase any brand Formally if a reservation utility of zero is assumed all consumers indexed on 0 z where consumer z z pAa is drawn in Figure 123 will not purchase any brand In this case the Page 313 Figure 123 Determination of the indifferent consumer among brands vertically differentiated on the basis of quality Left pA pB Right pA pB number of A buyers would be reduced to the size of the interval Exercise 1 in Section 129 addresses the case of reservation utility It is also clear from the fighthand side of Figure 123 that if the price of the lowerquality brand brand A is higher than the price of the highquality brand brand B pA pB then all consumers purchase only the highquality brand brand B For given locations of firms a and b in the second period each firm takes the price set by its rival firm as given and chooses its price to maximize its profit level Formally firm A and B solve After introducing all the assumptions for this model we now pause to give a precise definition for this twoperiod game We simply look for a subgame perfect equilibrium as described in Definition 210 on page 27 Definition 122 The quadruple ae be is said to be a vertically differentiated industry equilibrium if Second period For any given locations of firms a and b and constitute a Nash equilibrium First period Given the second periodprice functions of locations and ae be is a Nash equilibrium in location Page 314 Definition 122 is a subgame perfect equilibrium see Definition 210 on page 27 in which in the first period firms choose locations taking into account how their choice of location will affect the secondperiod equilibrium prices and hence profit levels It is important to note that the equilibrium actions of the firms in the second periods are functions not scalars of all possible given locations of firms We now proceed to solve the model starting from the second period The firstorder conditions to 125 are given by Hence Note that both equilibrium prices exceed marginal cost despite the fact that one firm produces inferior quality Equation 127 reveals that Proposition 122 The firm producing the higherquality brand charges a higher price even if the production cost for lowquality products is the same as the production cost of highquality products Substituting 127 into 125 yields that We now move to the first period where firm A takes be as given and maximizes πAabe given in 128 whereas firm B takes ae as given and maximizes πBae b It is easy to see that firm A would choose to produce the lowest possible quality and locate at ae 0 whereas firm B would choose to produce the highest possible quality and locate at be 1 This result is known as the principle of maximum differentiation Formally Proposition 123 In a vertically differentiated quality model each firm chooses maximum differentiation from its rival firm Are you confused Well you should be confused since in the horizontal differentiation model of subsection 731 we showed that when transportation costs are linear firms tend to move toward the center minimum differentiation However in a vertical quality differentiation Page 315 model the principle of maximum differentiation applies The reason for this difference is that in a vertically differentiated products model firms specialize in the production of quality for a certain consumer group Maximum differentiation implies that firms can increase their market power in their targeted consumer group 123 Market Structure Quality and Durability There is an extensive literature debating the relationship between the degree of a firms monopoly power and the quality or durability it chooses to build into a product see a survey article by Schmalensee 1979 That is the main question is whether a monopoly firm that is known to distort prices and quantity produced see chapter 5 also builds a shorter durability or a lower quality into its product than does a competitive industry Earlier writers on this subject Kleiman and Ophir 1966 and Levhari and Srinivasan 1969 concluded that firms with monopoly power have the incentives to produce goods of lower durability than would be produced by firms in a competitive market Contrary to this literature Swan 1970a b 1971 has demonstrated that there is actually no implied relationship between monopoly power and durability Swans novel result is known in the literature as the Swans independence result This result gave rise to an extensive literature examining the robustness of the independence result Levhari and Peles 1973 demonstrated that durability built in a product produced by a monopoly can be longer or shorter than under competition In addition they have shown that partial regulation of a monopoly that chooses strategies of quantity produced or price and durability or quality can reduce welfare where partial regulation is defined as a restriction by the regulating authority on either the quantity produced or the quality but not on both Kihlstrom and Levhari 1977 examine the robustness of Swans result by analyzing the effect of increasing returnstoscale IRS technologies on the production of durability Spence 1975 developed a fixedcost implying an IRS technology model to measure the divergence between the socially optimal quality level and the monopolys equilibrium quality level The debate on Swans independence result will probably continue forever However the reader is advised to learn the arguments given by the authors participating in this long debate In this section we provide a simple illustration of the Swans independence result by considering a monopoly firm selling light bulbs with variable durability Let us consider a consumer who lives for two periods Page 316 who desires light services for two periods Assume that the consumer is willing to pay an amount of V V 0 per each period of light services On the supply side assume that light bulbproducing firms possess the technology for producing two types of light bulbs a shortdurability light bulb yielding light services for one period only and a longdurability light bulb yielding light services for two periods The unit cost of producing the shortdurability light bulb is denoted by cS and the unit cost of producing a longdurability light bulb is denoted by cL where 0 cS V 0 cL 2V and cS cL For simplicity we ignore discounting and analyze market equilibria under extreme market structures monopoly and perfect competition Monopoly firm producing light bulbs The monopoly firm has the option of selling short or longdurability light bulbs and to charge a monopoly price for either type of bulbs First suppose that the monopoly sells shortdurability light bulbs Then since the consumer is willing to pay V per period of light services the monopoly would charge pS V per period and would sell two units one unit each period Hence the profit of a monopoly selling shortdurability light bulbs is given by Now suppose that the monopoly sells longdurability light bulbs Since the light bulb lasts for two periods the monopoly charges a price of pL 2V Hence the profit of the monopoly firm selling longdurability light bulbs is given by We would like to know under what condition the monopoly produces long or shortdurability light bulbs Clearly the monopoly produces shortdurability bulbs if Comparing 129 with 1210 yields Proposition 124 A monopoly producer of light bulbs would minimize the production cost per unit of duration of the light bulb Formally the monopoly would produce shortdurability light bulbs if 2cS cL and would produce longdurability bulbs if 2cS cL Proposition 124 illustrates Swans argument that despite the fact that them is only one seller the monopolys decision about which type of bulb to produce depends only on cost minimization and not on the market conditions such as the demand structure However to show Swans complete argument we investigate which type of light bulbs are produced in a competitive industry Page 317 Competitive light bulb industry Under perfect competition the price of each type of light bulb drops to its unit cost Hence pS cS and pL cL The consumer who desires two periods of light services would purchase a short duration light bulb If 2V pS 2V pL or if 2cS cL Similarly consumers purchase long durability light bulbs if 2V pL 2V pS or if cL 2cS Hence we can state Swans independent result by the following proposition Proposition 125 1 The durability of light bulbs is independent of the market structure 2 The firms would choose the level of durability that minimizes the production cost per unit of time of the products services It is important to note that this analysis assumes that our consumer is only concerned with the length of time service is provided by the product and does not attach any other value for durability per se This is rather an extreme assumption since if for example cost minimization yields the decision that light bulbs with durability of five minutes are produced then this means that our consumer has to replace a light bulb every five minutes Given that our consumer may attach value for the time it takes to buy and replace a light bulb it is unlikely that consumers will purchase shortduration light bulbs Similarly if cost minimization yields the decision that only singleshave razor blades are produced then consumers will have to buy a stock of 365 razor blades each year In this case it is clear that consumers would be willing to pay more than five times the amount they are willing to pay for a single shave blade for a fiveshave blade 124 The InnovationDurability Tradeoff All of us often wonder what to do with our old washing machine blackandwhite TV typewriter personal computer turntable or stereo When technologies keep changing rapidly consumers desire newtechnology products while they still receive some benefits from the oldertechnology product that they still own If all consumers have similar preferences and hence all desire the new technology products oldtechnology products cannot be sold in a market for used products Hence we sometimes get the feeling that with a rapidly changing technology goods are too durable That is we often say to ourselves some variation of My old computer does not want to break down so I dont know what to do with it once I replace it with a newer model Page 318 The question we investigate in this section is whether and under what conditions firms may produce products with excess durability from a social point of view In other words under what conditions do firms find it profitable to produce goods that will last for a very long time so that firms entering with new technologies will not be able to introduce and sell new products owing to the large existing supply of durable oldtechnology products This problem is analyzed in Fishman Gandal and Shy 1993 in an infinitehorizon overlapping generations framework Here we merely illustrate their argument in a twoperiod model with a simplifying assumption that in each period there is only one firm Consumers In period t 1 there is only one consumer who seeks to purchase computer services for the two periods of his or her fife t 1 2 In period t 2 one additional consumer enters the markets and seeks to purchase one period of the products services Let Vt denote the per period gain from the quality of the technology imbedded into the product a consumer purchases in period t and let pt be the corresponding price Altogether the per period utility of each consumer purchasing period t technology is Firms There are two firms Firm 1 operating in period 1 only is endowed with an old technology providing a per period quality level of vO to consumers Firm 2 a potential entrant in period 2 can produce the oldtechnology product vO however in addition firm 2 is endowed with the capability of upgrading the technology to a level of vN vN vO for an innovation cost of I 0 On the production side we assume that the production cost is independent of the technology level but depends on the durability built into the product Durability affects production costs since long lasting products are generally made with more expensive material say more metal relative to plastic cases and moving parts We say that the product is nondurable if it lasts for one period only That is a nondurable product is assumed to completely disintegrate after one period of usage We say that the product is durable if it lasts for two periods The unit production cost of a nondurable is denoted by cND whereas the Page 319 unit production cost of a durable is denoted by cD where we assume that cD cND That is we assume that durable goods are more costly to produce than nondurables With no loss of generality we also assume that the production of a nondurable product is zero cND 0 The twoperiod twofirm game is described as follows In period 1 firm 1 sells the oldtechnology product and therefore has to decide which price to charge p1 and whether to produce a durable D or a nondurable ND product In this second period firm 2 obviously chooses to produce a nondurable since the world ends at the end of period 2 and hence has to decide whether to invest in adopting the newer vN technology and the price p2 Figure 124 illustrates this twoperiod game Figure 124 Innovation and durability Below we analyze two situations based on whether firm 1 produces a durable or a nondurable in period 1 Secondperiod pricing given that firstperiod production is nondurable In the second period firm 2 offers either the oldtechnology vO product for sale or invests I for the adoption of its newtechnology vN product The pricing and innovation decision of firm 2 are summarized by That is when firm 1 produces a nondurable in period 1 then in period 2 both the old and the new consumers seek to purchase the product If the innovation cost is sufficiently low firm 2 invests in the improved technology and sells it to the old and new consumers However If I Page 320 is high firm 2 sells the old technology to both the old and the new consumers Secondperiod pricing given that the firstperiod product is durable Now suppose that firm I sells a durable in period 1 Then in period 2 the old consumer already possesses the vO technology product In this case firm 2 has two possibilities It can price its new technology product low enough at which induces the old consumer to discard his old technology durable and purchase the new product vN in this case Or it can price it high at so that only the new consumer purchases the newtechnology product while the old keeps using the old durable product In this case Comparing with yields Proposition 126 Suppose that firm I sells a durable to period 1 consumer Then in period 2 firm 2 sells the new technology product if In this case 1 if vN 2vO firm 2 sells its newtechnology product to both the old and new consumers 2 if vN 2vO firm 2 sells its newtechnology product to the new consumer only Firstperiod durability choice In period t 1 firm 1 chooses a price p1 and whether to produce a durable or a nondurable If firm I sells a nondurable 1211 implies that the maximum price firm 1 can charge for selling one period of the product service is In this case In contrast if firm 1 sells a durable 1211 implies that the maximum it can charge is given by since in this case the product provides a service of vO for two periods In this case Therefore comparing with yields Proposition 127 Firm 1 produces a durable if vO cD Otherwise it produces a nondurable Proposition 127 is rather simple Firm 1 would produce a durable if the extra profit from charging for secondperiod product service exceeds the difference in cost between producing a durable and a nondurable cD cND cD Page 321 Durability innovation and welfare We define the socialwelfare function as the sum of consumers utility levels and the firms profits over the two periods given by where and are the utility levels of period I consumer in periods 1 and 2 respectively U2 is the utility level of the consumer who lives in period 2 only and πt is the profit of the firm operating in period t We conclude from the previous analysis that there could be three types of equilibria 1 firm I produces a durable or a nondurable 2 firm 2 innovates and adopts the new technology or does not innovate 3 the combination of the two possibilities The type of equilibrium that obtains is determined by the exact parameter values In order to restrict the parameter range to interesting cases we assume that Assumption 121 vO cD and The first part of Assumption 121 implies that the firstperiod firm would find it profitable to produce a durable product The second part implies that the innovation cost for the new technology is at an intermediate range We now state our main proposition Proposition 128 Under Assumption 121 1 firm 1 produces a durable innovation will not occur and only the oldtechnology product will be sold and 2 this outcome is dominated from a socialwelfare viewpoint by an outcome where firm 1 produces a nondurable instead of a durable Proof Since vO cD Proposition 127 implies that firm I produces a durable in period 1 Now by way of contradiction suppose that firm 2 innovates Then if firm 2 sells to both consumers by Assumption 121 Similarly if firm 2 innovates and sells only to the young consumer also by Assumption 121 a contradiction Hence firm 2 will not innovate which proves part I of the proposition To prove part 2 we first calculate the social welfare under this outcome firm I produces a durable and firm 2 does not innovate In this case p1 2vO π1 2vO cD p2 vO π2 vO and Hence using 1213 Page 322 Now suppose that for some reason firm 1 is forced to produce a nondurable Then Assumption 121 implies that firm 2 does not innovate In this caseP1 vO π1 vO P2 vO π2 2vO and Hence using 1213 Comparing 1214 with 1215 implies that WND WD The intuition behind part 2 of Proposition 128 is as follows Durability in this model serves as a strategic means to capture future market share However durability per se does not serve any purpose to consumers and therefore to the social planner Since durability is costly to the economy the social planner can increase welfare by supplying a product of the same quality with no durability What policy conclusions can we derive from this model One recommendation would be for qualityregulating institutions such as standards institutes to allow short durability products into a market with rapidly changing technologies 125 The Market for Lemons So far we have analyzed markets where sellers could control the quality of the product they sell However there are many markets in which products with predetermined qualities are sold and therefore sellers are constrained to sell a product with a given quality If consumers can determine the precise quality by simply inspecting the product prior to the purchase if the product is a search good then the market will be characterized by a variety of qualities of the same product sold at different prices where higher quality brands will be sold for a higher price However in most cases buyers cannot determine the quality before the actual use the product is an experience good A natural question to ask is whether markets can function when buyers cannot observe qualities prior to purchase and when experience goods with different qualities are sold The reason the answer may be negative is that in such markets sellers need not adjust prices to reflect the actual quality of the specific product they sell In this section we analyze markets where sellers and buyers do not have the same amount of information about the product over which they transact That is we analyze markets with asymmetric information where sellers who own or use the product prior to the sale have a substantial amount of information concerning the particular product they own By contrast a buyer does not possess the knowledge about the quality of the particular product he wishes to purchase A second feature of the particular markets we analyze here is that Page 323 reputation does not play a role This assumption is unrealistic for certain markets where sellers generate most of their sales from returning customers In fact almost all the large retail stores in the United States are now allowing consumers to return the products for a full refund thereby guaranteeing satisfactory quality Reputation effects are also present in expensive restaurants where most sales are generated from fixed clientele Still there is a substantial number of markets in which reputation does not play a role For example our analysis will focus on the market for used cars Whether the seller is a private owner or a dealer the issue of reputation is of not of great interest to the seller Therefore if the seller possesses a lowquality product the seller has all the incentives to sell it as a highquality product The problem of asymmetric information between buyers and sellers is perhaps most noticeable in the market for used cars A buyer has a short time to inspect the car to check the engines compression and oil consumption and to perform other tests that can partially reveal the quality of the car Since full warranties are not observed in the market for used cars a buyer has to assume that with some probability the used car he buys may be a lemon Of course lemon cars need not be just old cars since all lemon cars have been initially sold as new cars However the difference between new lemon cars and used lemon cars is that the seller of a new car newcar dealer does not know the quality of the particular car he sells to a particular customer whereas a seller of a used car knows whether the particular car is a lemon or a good car Thus the markets for used and new cars have substantially different information structures 1251 A model of used and new car markets Following Akerlof 1970 let us consider an economy with four possible types of cars brandnew good cars brandnew lemon cars bad cars used good cars and used lemon cars All individuals in this economy have the same preferences for all the four types of cars We let NG value of a new good car NL value of a new lemon car UG value of a used good car and UL value of a used lemon car We make the following assumptions Assumption 122 1 The value of new and old lemon cars is zero that is NL UL 0 Page 324 2 Half of all cars new and old are lemons and half are good cars 3 New good cars are preferred over used good cars that is NG UG 0 The first and the second items of Assumption 122 are merely for the sake of simplifying the model The third item is clear and is intended to induce good usedcar owners to purchase new cars under certain price structure Assumption 122 implies that the expected values of new and used cars are given by Clearly the expected value of a new car exceeds the expected value of a used car EN EU There are four types of agents in this economy 1 new car dealers who sell new cars for an exogenously given uniform price denoted by pN Clearly since there is no knowledge of the quality of new cars all new cars are sold for the same price 2 individuals who do not own any car whom we call buyers in what follows 3 owners of good used cars whom we call sellers and 4 owners of lemon used cars whom we also call sellers We denote by pU the price of a used car Since usedcar buyers cannot distinguish between lemon used cars and good used cars all used cars are sold for the same price pU We assume that each buyer maximizes the expected value of a car minus a price in case the agent is a buyer Formally the utility of a buyer who does not own any car is assumed to be The utility of a seller of a good used car who sells his used car for pU and buys a new car for pN is given by Finally the utility of a seller of a lemon car who sells his used lemon for pU and buys a new car for pN is given by Page 325 That is each usedcar owner has the option to maintain his car thereby gaining a utility of UG or UL depending on whether he owns a good or a lemon used car or to buy a new car for pN and in addition get paid pU for selling his used car The problem of the buyers The buyers do not own any car and therefore have the option of either buying a new car or buying an old car Thus in view of 1217 buyers will buy a used car if or if pU satisfies The problem of the lemon usedcar seller An owner of lemon used car has the option of keeping his car gaining zero utility or selling his used car and buying a new car In view of 1219 an owner of a lemon used car sells his car if or The problem of the good usedcar seller An owner of a good used car has the option of keeping his car or selling his used car and buying a new car In view of 1218 an owner of a good used car sells his car if or Figure 125 summarizes the cases given in 1220 1221 and 1222 in the pN pU space where used cars are either demanded or offered for sale The two regions of interest are the upper one corresponding to 1222 where pU is sufficiently high so that an owner of a good used car offers his or her car for sale and the lower region corresponding to 1220 where buyers those who do not own cars find pU to be sufficiently low and decide to purchase a used car Figure 125 shows that the combinations of pN and pU satisfying the condition in which good used cars are sold do not satisfy the condition in which buyers would demand used cars That is the region in which pU is high enough to induce an owner of a good used car to sell his or her good car does not intersect with the region in which pU is low enough to induce a buyer to purchase a used car instead of a new car This proves our main proposition known as the Lemons Theorem Proposition 129 Good used cars are never sold That is lemon used cars drive good used cars out of the market Page 326 Figure 125 The market for lemons Bad cars drive out the good cars The prices of new and used cars corresponding to cases where used cars are demanded or offered for sale I Good usedcar seller sells II Bad usedcar seller sells III Buyers demand used cars Note The Figure assumes UG NG2 A reader who may have purchased a good used car may wonder how it happened That is we sometimes observe that good used cars are sold in the market despite what Proposition 129 predicts The reason this happens follows from our assumption that usedcar owners may sell their cars only if they wish to buy a new one However it often happens that usedcars owners sell their cars for different reasons such as moving to another state or abroad Thus the observation that sometimes good used cars are sold does not contradict Proposition 129 1252 Applications of the lemon problem The model described in the previous subsection can be applied to describe a wide variety of other markets as well Consider the health insurance market where both healthy and sick people wish to purchase health insurance from an insurance company or an HMO The buyer of an insurance policy knows whether he is healthy However the insurance company has no prior information on the particular buyer unless it requires that all buyers go through an extensive medical checkup If the insurance price reflects the average treatment costs for a certain period then by the same argument as in the previous subsection it is clear that only sick people would purchase health insurance So the remaining question is how can insurance companies or HMOs make a profit The answer is probably that insurance companies attempt to discriminate on Page 327 the basis of price charge different rates according to age and according to health problems the patient had prior to filing the insurance applications A similar problem occurs in other insurance coverage namely risky drivers tend to buy extended coverage for their car Insurance companies can partially solve this problem by charging different rates according to age location and distance to work following the data they collect on accident frequencies Consider now the market for allyoucaneat restaurants The buyers are divided into two groups of people those who eat a lot and those who eat very little If the price of a meal reflects the food cost of the average eaters then it is clear that only very hungry people would go to allyoucaneat restaurants whereas less hungry people would generally prefer to pay for each specific dish they order The question is then how can allyoucaneat restaurants earn a profit The answer is perhaps that many allyoucaneat restaurants also serve regular meals and as with most restaurants they earn the profit on side dishes such as drinks and desserts Consider now a labor market in which firms cannot distinguish between productive workers and lazy ones If the ongoing market wage reflects the average productivity of a worker it is clear that a good worker who has an alternative wage which does reflect his productivity will not apply for a job at the ongoing wage rate Thus the lemon theorem suggests that only less productive workers apply for jobs Spence 1974 suggests that good workers may take some acts that will distinguish them from the less productive workers thereby signaling to the firms that they are productive signaling is discussed in subsections 846 126 and 127 One act would be to go to college Although college does not necessarily improve the skill of the worker going to college may signal to firms that the graduate is a productive worker since unproductive workers may not be able to graduate and therefore would not benefit from investing in education 126 QualitySignaling Games Consumers are often unable to recognize the quality of a product before they actually purchase and use the product even if they are aware that both highquality and lowquality brands are sold in the market We refer to such goods as experience goods Producers however have more information regarding the brands they sell and in most cases are fully aware of their product This creates a problem of asymmetric information that we first analyzed in subsection 846 In that section we analyzed an entrydeterrence problem in which the potential entrant did not know the production cost of the incumbent and the incumbent had Page 328 to signal its production cost by the price it charged prior to the threat of entry In this section we analyze a technically similar problem in which a monopoly firm knows the quality of the brand it sells but consumers are unable to learn the brands quality prior to the actual purchase Our goal is to demonstrate that a monopoly firm can signal the quality it sells by choosing a certain price and by imposing a quantity restriction on the brand it sells We should note that signaling models are derived from Spence 1974 for an application to quality signaling see Wolinsky 1983 Suppose that there is a continuum of identical consumers With no loss of generality we normalize the number of consumers to equal 1 Each consumer buys at most one unit and knows that the product can be produced in two quality levels high k H and low k L where H L 0 For a given price denoted by p the utility function of each consumer is given by Suppose that each consumer goes to the monopolys store and observes a price level of p dollars Will consumers purchase the product if they find that p H Clearly not since there is a possibility that the product may be of low quality and in this case 1223 implies that such a purchase results in a utility level below zero which is the reservation utility level We now describe the monopoly producer side Denote by cH the unit production cost of the monopoly if it is a highquality producer and by cL if it is a lowquality one where That is the unit production cost of a highquality product exceeds that of a lowquality product We make the following assumptions Assumption 123 1 The monopolist is a highquality producer 2 Production costs are sufficiently low relative to consumers valuation of the two qualities Formally L cH The second part of Assumption 123 ensures that a highquality producer can charge p L for a highquality product without making a loss We assume that the strategy available to the monopolist is twodimensional so that it can choose the price p and the quantity produced q Clearly since the total number of consumers is Page 329 normalized to equal 1 We wish to solve the problem how a highquality monopolist can sell a highquality product given that consumers are not sure whether the brand they buy is a highquality one In other words how can a highquality producer convince the consumers that he or she does not cheat by selling a lowquality brand for a high price Hence in choosing the price and quantity levels the producer needs to signal his or her high quality to the consumer Proposition 1210 There exists a pair of a price and a quantity level that convinces consumers beyond all doubts that the brand they buy is a highquality one Formally if the monopolist sets then a consumers can infer that the brand is of high quality b qm consumers will purchase the product and 1 qm consumers will not purchase the brand due to the lack of supply Before proving this proposition we think it is worthwhile to repeat that the essence of signaling is the firms to choosing a pricequantity combination that would signal to the consumer that the product is of high quality In order to do that the monopoly must choose both a price and a quantity produced that a low quality producer would not find profitable to set Using this action the monopoly can convince the consumer that it is not a lowquality producer Proof The monopoly has to show that a lowquality producer would not choose pm and qm as the profitmaximizing price and quantity If the monopolist were a lowquality producer then he or she could clearly sell to all consumers for the price p L and make a profit of πLL 1 1L cL Let us note that this profit level is attainable by a lowquality producer Clearly at this price all consumers would purchase the product Now the question is whether a lowquality monopoly could profitably choose pm and qm as the profitmaximizing price and quantity Suppose it does Then Thus using these price and quantity levels a highquality monopolist is able to demonstrate that had he or she been a lowquality producer he or she could earn the same profit by setting p L and selling to all consumers instead of setting pm and qm That is by cutting the profit level to that of what a lowquality monopolist could collect under perfect Page 330 information the highquality producer convinces the consumers that he or she is not a lowquality one since if he or she were a lowquality producer he or she could make the same profit level So far we have showed that a highquality producer can signal his or her quality level to the consumer by using the price and quantity instruments so that consumers uncertainty is completely resolved However this signaling mechanism raises two questions What is the cost paid by the monopoly to resolve consumers uncertainty Would this highquality monopoly find it profitable to signal its high quality level to the consumers To answer these questions we need to calculate the profit level of a highquality monopolist when he or she sets pm and qm Hence Hence comparing this profit level to the profit under perfect information H cH yields the cost of revealing information The answer to our second question depends on whether Crossmultiplying 1224 yields that this inequality always holds since H L cH cL Criticism of the qualitysignaling model The qualitysignaling model developed in this section is used only for the sake of illustration Note that if a firm can choose whether to become a lowquality or a highquality producer it would choose to be a lowquality producer That is since a highquality producer needs to signal his quality and since production cost is higher it becomes more profitable to be a lowcost producer This model can be modified to capture profitable signaling by adding consumers who purchase only high quality goods 127 Warranties One common method of insuring the consumer against defects in the product is to bundle the product with a warranty There are many kinds of warranties Some warranties restrict the manufacturers liability only to parts others to labor and parts in case that repair is needed Most warranties are limited to a certain time period after the purchase whereas few provide a lifetime warranty We shall not discuss in the Page 331 present section why most warranties are limited The reason has to do with the moral hazard phenomenon a situation where a full warranty will provide the consumer with the incentives to misuse the product or not to take proper care or it see Cooper and Ross 1985 Therefore in order to demonstrate the role of warranty in market behavior we make the following assumption Assumption 124 1 The product can be either fully operative or fully defective A defective product has no value to the buyer and cannot be resold for scrap 2 At the time of purchase neither sellers nor buyers know whether the specific product is defective 3 The manufacturerseller has two options regarding the sale of the product a He or she can sell the product without a warranty In this case if the specific product is found to be defective the buyer loses the entire value of the product b He or she can sell the product with a full replacement warranty which guarantees full replacement of a defective product with no loss of value to the buyer That is if the replacement product is also found to be defective the monopoly is obligated to replace the replacement product and so on In the literature Grossman 1980 provides a comprehensive analysis of a monopoly that can offer a warranty for the product it sells Spence 1977 builds on a signaling argument and shows that higherquality firms offer a larger warranty than do lowquality firms In what follows we confine our analysis to a monopoly selling a product to a competitive consumer where the product has a certain probability of being defective The next subsection discusses the monopoly optimal provision of warranty under symmetric information between the buyer and the seller A subsequent subsection analyses a market in which warranties can serve as a partial signal of the products quality 1271 Warranties under symmetric information Consider a product whose value to the consumer is V if the product is operative and 0 if the product is defective where V 0 Suppose that there is a known probability for products of this type to be functional We denote this probability by ρ where 0 ρ 1 Thus with probability Page 332 1 ρ the product produced by the monopoly will be found to be defective In this subsection we assume that the seller and the buyer have symmetric information regarding the products reliability meaning that both the seller and the buyer know the product is reliable with an exogenously given probability ρ Let p denote the monopoly price and c 0 denote the unit production cost of the product We assume that the utility function of the consumer is the expected value of the product minus the products price if he buys the product and zero if he does not buy the product Formally Finally we assume that ρV c which implies that the expected utility from the product exceeds the unit production cost Assuming otherwise would yield that the product will not be produced since the monopoly will not be able to get consumers to pay a price exceeding unit cost The profitmaximizing monopoly has the option of selling the product with or without a warranty No warranty With no warranty 1225 implies that the maximum price the monopoly can charge is the expected value of the product Thus if we assume one consumer then under no warranty the monopoly price and profit level are given by Warranty When the monopoly provides the consumer with a full replacement warranty under Assumption 124 the consumer is assured of gaining a value of V from the product We need the following Lemma Lemma 121 The expected unit production cost for a firm providing a full replacement warranty is cρ Proof The cost of producing the product is c If the product is defective expected cost increases by 1 ρc If the replacement product is defective then expected cost increases again by 1 ρ2c and so on Hence expected cost is given by Page 333 Thus Lemma 121 implies that the expected production cost is c when zero failure probability and becomes infinite as since in this case the product is produced and replaced infinitely many times Altogether the maximum price a monopoly can charge and the profit level are given by Will the monopoly sell with a warranty Comparing 1226 with 1228 yields the conclusion that πW πNW if V cρ which must hold for the monopoly to make profit under any warranty policy Hence we can conclude the analysis with the following proposition Proposition 1211 Under symmetric information where the reliability parameter p is common knowledge a monopoly will always sell the product with a warranty The intuition behind Proposition 1211 is as follows When the monopoly provides a warranty the monopoly can increase the price by 1 ρV above the price selling with no warranty The associated increase in cost is by assumption Hence by providing a warranty and given that the monopoly extracts all consumer surplus the monopoly can increase its price by more than its increase in the cost associated with replacing the products with a certain probability of failure In other words consumers are willing to pay more for the warranty than what it costs the seller 1272 The role of warranties under asymmetric information In section 125 we encountered the problem of asymmetric information between sellers and buyers where we assumed that sellers are generally better informed about the products quality than the buyers Since consumers are not informed they cannot distinguish between highly reliable products products with a high probability of not breaking down and products with a high defective rate In this subsection we continue with the exploration of markets with asymmetric information and analyze a duopoly in which one firm produces a reliable product high probability of being operative and one firm produces an unreliable product with a low probability of being operative However the consumer does not have any way of knowing Page 334 which one of the firms produces the more reliable product That is the consumer cannot distinguish between the two products We show that by providing a warranty with the product and choosing a certain price the highquality firm can signal to the consumer that it is selling the more reliable product In this case the consumer can conclude beyond all doubt that the highquality firm is indeed a highquality producer and not a lowquality producer masquerading as a highquality firm The signaling principle always remains the same if a highquality producer wants to prove to the consumer that he or she is a highquality producer he or she has to carry an act that is unprofitable for a lowquality producer From this act the consumer will conclude that the producer does produce a highquality product and will be willing to pay for the product accordingly Consider an economy with two producers A highquality producer selling a product with probability ρH of being operative and a lowquality producer producing a product with probability ρL of being reliable 0 ρL ρH 1 No warranties Since the consumer cannot distinguish between the producers before the purchase both products high and low reliability are sold for the same price In this case since from the consumers point of view the products are homogeneous before the purchase a Bertrand price competition see section 63 leads to a unique equilibrium where prices equal the unit cost hence zero profits That is pNW c and i H L Therefore with equal production cost both high and lowquality products are produced and the high quality manufacturer cannot be identified by the consumer Warranty as a signal We now show that by providing a warranty and choosing an appropriate price the highquality producer can signal to the consumer that he or she sells a reliable product Proposition 1212 Let V c The highquality producer can push the low quality producer out of the market by setting pw cρL and providing a warranty In this case the consumer will buy only the more reliable product and the highquality producer will make a strictly positive profit Proof We first show that a lowquality producer will not find it profitable to sell his or her product with a warranty at this price To see Page 335 that using 1227 we calculate that This concludes the main part of the proof To complete the proof we need to verify that first the consumer will indeed prefer purchasing the more reliable product with a warranty instead of the less reliable product at the lowest possible price p c and second the highquality producer makes an above zero profit To see this observe that the profit of the highquality firm is given by Finally the utility of a consumer buying the more reliable product exceeds the utility of buying the less reliable product without a warranty even if the less reliable product has the lowest possible price c since 128 Appendix The Legal Approach to Products Liability In this section we briefly describe the legal approach to products liability which is concerned with defective products and trades The reader interested in learning all the legal issues concerning liability should consuit Howard 1983 and Phillips 1988 for a comprehensive analysis of product liability Liability refers to the obligation of the producer or the merchant seller to those who were damaged as a result of a defective product Note that those damaged need not be only the buyers but could also be bystanders and owners of property 1281 Defects and liability In general there are four types of defects production defects design defects erroneous operating instructions and warnings and mislabeling and misrepresentations of products Thus liability law extends the liability beyond what are purely understood as manufacturing flaws Clearly these distinctions are hard to make but they seem to be important in deciding what standard of liability strict liability or negligence is assumed for the manufacturer For example it seems more likely that Page 336 strict liability is generally imposed for production defects than for design defects Misrepresentation defects may or may not be judged under strict liability Under these classifications it is necessary to determine whether the product is defective The most common way to make that determination is to rely on consumer expectations meaning that the product sold must be more dangerous than the ordinary consumer with the ordinary knowledge common to the community would expect it to be A problem may arise when the consumer buys products known to be dangerous since an ordinary consumer should expect the danger associates with this product Another way of determining defectiveness is to ask whether the seller would have sold the product had he or she known the potential harm resulting from the sale Thus in this case defectiveness is defined as a presumed knowledge by the seller about the quality of the product Defectiveness can also be determined by determining whether the producer invested a sufficient amount in preventing a risk where sufficiency estimated by balancing the cost of preventive investment and the monetary value of the inflicted damage or risk caused by the product in the condition it was sold Liability is not limited solely to the producer Liability may be assumed to rest on any commercial seller such as dealers vendors constructors stores and so on However strict liability is less likely to be imposed on them since the presumed knowledge of the seller is smaller than that of the maker of the product 1282 Warranties The Uniform Commercial Code states that unless excluded or modified a warranty is implied in the contract of sale However a warranty is not implied if the seller is not a merchant The implied warranty which attaches strict liability to the seller is important since it reduces the chances that written agreements such as warranty certificates or disclaimers would always be effective in reducing the sellers liability In order for the seller to reduce his or her liability to a level below that assumed in the implied warranty he or she has to provide a disclaimer however a disclaimer is not always accepted by courts A disclaimer is generally accepted in the case of negligence on the part of the consumer Also a disclaimer is valid only with respect to the trading parties buyers not for example with respect to bystanders Since warranties have been recognized as a special source of the deception of consumers the FTC has issued several rules some of which have been adopted as laws that require that the terms of the guarantee will be clear and presented in a clear fashion All of us who have been Page 337 given warranties can imagine that the amount of information that has to be included in a warranty must be enormous For example what is the time interval corresponding to a life time warranty that we often see on back of our packages What is meant by full warranty Does a full warranty include labor cost parts freight or the loss of time associated with the loss of use 129 Exercises 1 Consider the modified Hotelling verticaldifferentiation model of subsection 1222 but suppose that consumers have a reservation utility in the sense that a consumer prefers not to buy any brand if his or her utility falls below zero Recall that the preferences exhibited in 122 imply that there is no lower bound on utility from consumption Figure 123 implies that this modification in preferences would not affect the number of highqualitybrand buyers since all consumers indexed on gain a strictly positive utility from buying the highquality brand However point z in Figure 123 shows that no consumers indexed on 0 z will purchase any brand since otherwise their utility falls below zero Perform the following exercises a Show that for given a b pA and pB the number of consumers who do not purchase any brand equals to z pAa b Conclude that the market share of firm A is c Using the same procedure as in 125 show that for given a and b the secondperiod equilibrium prices and profit levels are given by d Show that in the first period firm A would choose to locate at ae 47 whereas firm B would locate at be 1 2 Prove the second part of Proposition 121 using the same procedure as the one used in the proof of the first part 3 Consider the lemon model described in section 125 and suppose that the owner of the good used car must sell his or her car because he or she is leaving the country Assume that the market prices of used and new cars are exogenously given by and respectively Characterize the demand and supply patterns of the four types of agents under these prices Page 338 4 Consider the monopolys warranty problem under symmetric information analyzed in subsection 1271 but assume that for some reason the monopoly cannot guarantee more than one product replacement in case the product purchased is found defective That is if the product is found defective the monopoly can provide a warranty to replace the product with a new product however if the replacement product fails then the monopoly cannot replace the replacement product a What is the monopolys expected cost if it provides this type of warranty b What is the maximum price the monopoly can charge for the product sold with this type of warranty c Conclude whether Proposition 1211 holds for this type of warranty 1210 References Akerlof G 1970 The Market for Lemons Qualitative Uncertainty and the Market Mechanism Quarterly Journal of Economics 89 488500 Beath J and Y Katsoulacos 1991 The Economic Theory of Product Differentiation Cambridge Cambridge University Press Cooper R and T Ross 1985 Product Warranties and Double Moral Hazard Rand Journal of Economics 16 103113 Fishman A N Gandal and O Shy 1993 Planned Obsolescence as an Engine of Technological Progress Journal of Industrial Economics 41 361370 Gabszewicz J and J Thisse 1979 Price Competition Quality and Income Disparities Journal of Economic Theory 20 340359 Gabszewicz J and J Thisse 1980 Entry and Exit in a Differentiated Industry Journal of Economic Theory 22 327338 Grossman S 1980 The Role of Warranties and Private Disclosure about Product Quality Journal of Law and Economics 24 461483 Howard M 1983 Antitrust and Trade Regulation Selected Issues and Case Studies Englewood Cliffs NJ PrenticeHall Kihlstrom R and D Levhari 1977 Quality Regulation Efficiency KYKLOS 30 214234 Kleiman E and T Ophir 1966 The Durability of Durable Goods Review of Economic Studies 33 165178 Levhari D and Y Peles 1973 Market Structure Quality and Durability Bell Journal of Economics 4 235248 Levhari D and T N Srinivasan 1969 Durability of Consumption Goods Competition versus Monopoly American Economic Review 59 102107 Page 339 Oi W 1973 The Economics of Product Safety Bell Journal of Economics 4 328 Phillips J 1988 Products Liability in a Nutshell 3rd ed St Paul Minn West Publishing Co Phlips L and J Thisse 1982 Spatial Competition and the Theory of Differentiated Products An Introduction Journal of Industrial Economics 31 111 Schmalensee R 1979 Market Structure Durability and Quality A Selective Survey Economic Inquiry 17 177196 Shaked A and J Sutton 1982 Relaxing Price Competition Through Product Differentiation Review of Economic Studies 49 113 Spence M 1974 Market Signaling Cambridge Mass Harvard University Press Spence M 1975 Monopoly Quality and Regulation Bell Journal of Economics 6 417429 Spence M 1977 Consumer Misperceptions Product Failure and Producer Liability Review of Economic Studies 44 561572 Swan P 1970 Durability of Consumer Goods American Economic Review 60 884894 Swan P 1970 Market Structure and Technological Progress The Influence of monopoly on Product Innovation Quarterly Journal of Economics 84 627638 Swan P 1971 The Durability of Consumer Goods and the Regulation of Monopoly Bell Journal of Economics 2 347357 Wolinsky A 1983 Prices as Signals of Product Quality Review of Economic Studies 50 647 658 Page 341 Chapter 13 Pricing Tactics TwoPart Tariff and PeakLoad Pricing People want economy and theyll pay any price to get it Attributed to Lee Iacocca Youd be surprised how much it costs to look this cheap Attributed to Dolly Parton The pricing techniques discussed in this chapter are generally studied under the subject of public utility pricing where a regulating agency such as the state city or any other local government controls the prices and quality of service provided by the public utility However as the reader will discover these pricing techniques are also used by unregulated and privately owned firms The major difference between regulated publicutility pricing and prices chosen by privately owned firms is that a regulator attempts to choose prices intended to maximize consumer welfare whereas unregulated firms choose prices to maximize profit As it turns out in many cases the regulator and an unregulated monopoly will choose to set similar price structures that may differ only by a lump sum transfer from consumers to firms In what follows we study several pricing techniques employed by unregulated profitmaximizing firms Section 131 TwoPart Tariff analyzes why sports clubs tend to charge annual membership fees instead of or in addition to fixing a price per visit Twopart tariffs are also charged by some cable TV companies and by wholesale club stores Section 132 Nonuniform Pricing generalizes the twopart tariff Page 342 to the case of heterogeneous consumers and demonstrates how quantity discounts can increase firms profit by extracting higher surplus from different consumer groups Section 133 PeakLoad Pricing analyzes firms choices of capacity and prices when the demand is seasonal for example the choices of airline firms car rental companies hotels resorts regulated and unregulated phone and electricity companies universities day versus evening classes movie theaters restaurants and many others Section 134 Can Firms Control the Seasons concludes with an extension of the peakload pricing problem by having firms set prices to manipulate the relative quantity demanded between seasons 131 TwoPart Tariff It has been observed that many commercial enterprises charge annual membership dues instead of or in addition to pricing each unit of consumption separately This phenomenon is observed mostly in entertainment industriessuch as amusement parks most sports clubs and some theatersand recently in wholesale clubs Oi 1971 proposed an explanation for this observation Given downwardsloping demand when a monopoly charges a fixed price per unit of consumption if consumers purchase the product then they gain positive consumer surplus see subsection 323 on page 52 Thus even when a monopoly charges its profitmaximizing price it is unable to extract the entire consumer surplus Therefore in addition to the per unit price a monopoly firm needs to set a second pricing instrument in order to be able to extract the entire consumer surplus 1311 Clubvisiting consumers Suppose that a consumer gains satisfaction from club visits and from other goods which we term as money We denote by Q the number of club visits and by m the amount of money spent on other goods Let the consumer earn a fixed income of I to be spent entirely on club visits and other goods We denote by φ the membership dues and by p the price per visit Thus the consumers budget constraint is given by The utility of our funloving consumer is a function of the number of club visits Q and the consumption of other goods m We assume a quasilinear utility function given by Page 343 Figure 131 illustrates a set of indifference curves derived from this utility function In Figure 131 the indifference curve U0 originating Figure 131 Quasilinear utility indifference curves from the income level I is associated with the initial utility from spending all the income I on other goods This indifference curve shows the combinations of club visits and spending on other goods that leave the consumer neither better off nor worse off than spending all the income on other goods We now derive the consumers demand curve for club visits Substituting 131 into 132 for m yields the consumerutilitymaximization problem Hence for given p and φ at a sufficiently low level the consumer chooses the number of visits Q that solves yielding a demand function 1312 No club annual membership dues Suppose that the club has a limited capacity Formally assume that the clubs capacity is limited to K visitors K 0 We now suppose that the club has only one method of collecting money from the club visitors which is charging a price p per visit where club membership is not required That is the club sets the annual membership dues to φ 0 Page 344 When φ 0 the monopoly club chooses Q to maximize Proposition 131 Under the preferences given in 132 the monopoly club sets the price so that the demand for club visits equals its capacity Formally Proof The preferences 132 yield an elastic demand curve 134 implying that the clubs profit rises with the number of visits Therefore the club will operate under full capacity The consumption point is illustrated in Figure 131 at the point E1 where the price line budget constraint is tangent to the indifference curve labeled U1 U1 U0 Hence under a pricepervisit structure with no membership fees the welfare of the consumer must increase compared with the noclubvisits allocation As we show below this is not necessarily the case when club charges involve annual fixed dues 1313 Annual membership dues Annual membership fees fixedpart tariff is in fact a bundling method discussed in section 141 in which the club offers the consumer the opportunity to pay a fixed amount of φ 0 and to receive a package containing a fixed number of free visits Figure 131 shows that for a package containing Q K number of visits the consumer is willing to pay a maximum amount of φ2 That is consuming a package of K visits for a fixed fee of would leave the consumer no worse off than he or she would be with the noclubvisits case We now calculate the maximum annual fee that the club can charge for K number of visits that make it worthwhile for the consumer to purchase To do that let us observe that in Figure 131 by construction the point E2 lies on the initial indifference curve U0 That is the club sets φ just about the level where the consumer is neither better off nor worse off by joining the club Formally the club sets φ2 that solves implying that and hence Hence Page 345 Proposition 132 A fixed fee for a bundle of visits yields a higher profit to the club than any profit generated with a per unit price with no annual fee Formally 1314 Twopart tariff In practice a club would hesitate charging exactly φ2 as a membership fee mainly because a small mistake in estimating the exact location of the indifference curve U0 or the consumers income may result in no sales at all A second reason why a firm would not use only a fixed membership fee is that consumers may have heterogeneous preferences so that a high membership fee may induce only a partial participation We therefore conclude that clubs would generally charge a lower fee than the maximum fee calculated in the earlier subsection For example Figure 132 demonstrates a possible package of Q3 club visits for an annual fee equal to φ3 Clearly the consumer buys such a package Figure 132 Pure twopart tariff club charges However Figure 132 also shows that the club can further increase its profit by supplementing the membership fee φ3 with an option to purchase additional visits for a price of p3 per unit In this case for P3 that is not too high the consumer purchases additional visits bringing the total number of visits to Q4 as illustrated in Figure 132 Page 346 132 Nonuniform Pricing Section 131 demonstrated how a twopart tariff can increase firms profit above the monopolys per unitprice profit level by employing two price instruments the conventional per unit price and the lumpsum consumption independent fixed membership dues The profit gains from using the two part tariff are due to the monopolys ability to extract higher surplus from a given group of homogeneous consumers In this section we demonstrate a price strategy commonly used by rums to price discriminate among heterogeneous groups of consumers The nonuniform price schedule is a tariff for one or more goods in which the consumers total outlay does not simply rise proportionately with the amounts of goods the consumer purchases That is a nonuniform price schedule consists of quantity discounts and quantity premiums for extensive analysis of nonuniform pricing see Brown and Sibley 1986 Figure 133 illustrates the inverse demand for local phone calls by two different groups households and business given by pH 122qH and PB 6qB2 respectively where prices are given in cents Assuming zero marginal cost in providing phone services section 53 Figure 133 Nonuniform pricing and price discrimination on page 75 shows that a monopoly selling in two segmented markets markets in which arbitrage cannot take place would set quantity produced in each market by equating MRHQH MCQH QB MRBQB 0 thereby charging different prices in the two markets given by pH 6 and pB 3 and producing qH 3 and qB 6 There Page 347 fore if the monopoly can price discriminate it would charge business lower rates than it would charge households for local phone calls The problem facing the monopoly is how to set the price schedule in a way that would induce the two different groups of consumers to pay different prices and to consume different quantities In general there are many reasons why a firm may not be able to charge different prices to different groups of consumers for example price discrimination is illegal under the Clayton Act see subsection 563 also a monopoly may not be able to identify the consumers belonging to a particular group Altogether we now demonstrate that nonuniform pricing can generate the price discrimination monopoly outcome even when the monopoly does not directly discriminate among the different groups of consumers or cannot simply identify these groups We now investigate the price schedule illustrated in Figure 134 Figure 134 Nonuniform price schedule Proposition 133 Consider the pricepercall schedule illustrated in Figure 134 and formally given by Regular Rate Program Pay 6 cents per phone call Quantity Discount Program Pay a reduced rate of 3 cents per phone call but be charged for at least 9 phone calls Then this price schedule yields the same market prices as those charged by a discriminating monopoly Proof Clearly Figure 133 implies that when pH 6 households demand QH 3 phone calls and given pB 3 business customers demand QB 6 phone calls We need to show to show that households will not benefit from adopting the quantitydiscount price scheme If households adopt the regular rate their consumer surplus subsection 323 on page 52 is CS6 6 32 9 Page 348 If households adopt the discount rate then they are forced to buy 9 phone calls and actually use only 6 which makes the grossconsumer surplus equals the entire area under the demand curve given by 12 62 Since households are required to pay for 9 phone calls their net consumer surplus is Given that the households are indifferent between the two plans we can assume that they do not purchase the discount plan Clearly when p 6 businesses will purchase zero on the regular payment program However when they choose the discount plan Hence businesses will choose the discount plan Finally it can be shown that this monopoly phone company makes a higher profit under nonuniform pricing than under uniform pricing 133 PeakLoad Pricing The problem of peakload pricing is generally studied in the context of optimal governmental regulations for public companies such as public utilities including phone transportation and electricity companies see Brown and Sibley 1986 Joskow 1976 Sherman 1989 and Steiner 1957 However it should be emphasized that unregulated firms also benefit from setting peakload pricing simply because peakload pricing tends to be efficient and profitable when demand is periodic and when the investment in capacity is irrevocable in the short run For example private firms such as hotels restaurants sports clubs movie theaters and airlines and other transportation companies are all subject to seasonal demand schedules that vary between yearly seasons days of the week or the hours of the day We therefore focus our analysis on a privatesector monopoly firm which could represent an airline a hotel or a restaurant and then conclude with a discussion on the role of the regulator in controlling the prices Three factors characterize the peakload pricing problem First the levels at which demand varies between periods Second capital has to be rented or leased for a long period That is since the firm must commit in advance to the level of the plants capacity and since this commitment cannot be reversed between periods the duration of these contracts affect firms seasonal pricing decisions Third the firms output products or services is too costly or impossible to store Otherwise if the output is storable then the firm could produce equal amounts in each period Page 349 or all the output in a single period and then allocate the output across periods according to demands Consider a monopoly airline company flying on a single route during high H and low L seasons 1331 Seasonal passengers We let pH QH pL and QL denote the price and quantity of tickets in the high and low seasons respectively The demand for flights in each season is given by Figure 135 illustrates the seasonal demand structure Figure 135 Seasonal demand structure and monopoly peakload pricing 1332 Seating capacity and the airlines cost structure The monopoly airline faces two types of costs Capacity cost which is the number of airplane seats the airline rents for the entire year and variable cost which is the cost associated with handling each passenger which includes checkin luggage and food services For simplicity we ignore other costs commonly associated with airline operations such as airport charges see section 172 of an analysis of the airline industry We denote by r r 0 the unit capacity cost Thus if the airline rents aircraft capacity that can fly K passengers throughout the year its total capacity investment cost is rK We denote by c the operational Page 350 variable cost per passenger Thus assuming that seating capacity cannot be rented for less than one year high and low seasons together the airlines total cost when it flies QH passengers in the high season and QL in the low season is Equation 138 highlights the difference between a twomarket discriminating monopoly analyzed in section 53 and the present problem in which a monopoly airline faces two independent seasonal markets The difference between the analysis of section 53 and this problem follows from the fact that investment in capacity for the high season implies that no investment in seating capacity is needed for the low season Thus 138 implies that the airline monopoly cost structure exhibits joint production where production cost in one market also partially covers the cost of producing in a different market different season 1333 Profitmaximizing seasonal airfare structure In section 53 we proved that a monopoly discriminating between markets determines the price charged and quantity produced for each market by equating the marginal revenue in each market to its marginal cost However how should we calculate the airlines marginal cost in the present case Clearly the operational cost c is part of the unit cost but how do we allocate the unitcapacity cost between the markets The following proposition assumes that the lowseason demand is significantly lower than the highseason demand see Steiner 1957 Proposition 134 The monopolys profitmaximizing seasonal pricing and output structure is determined by That is capacity is determined only by the highseason demand where the highseason marginal revenue equals the sum of the operational and capacity marginal costs Proof Clearly given the linear shift of demand between the seasons the profitmaximizing output levels satisfy QH QL Hence meaning that in the low season the airline does not fly at full capacity Consequently the marginal cost of flying one additional passenger in the low season is independent of k Hence according to section 53 Page 351 the profitmaximizing lowseason number of serviced passengers is determined by is MRLQL c Therefore investment in capacity is determined only by the highseason demand so if we follow section 53 the monopoly sets is MRHQH c r 1334 Peakload pricing and efficiency Many utility companies gas local phone electricity and transportation are regulated in most states and they have to adhere to price schedules determined by the corresponding government Most states require that utility companies especially electricity submit variableload price structures based on the efficient marginalcost pricing principle If we move to the regulators problem we discover that the fact that the monopoly faces periodic demand schedules does not complicate the problem beyond the regulators problem when the monopoly faces a stable demand Thus given that marginalcost pricing is efficient Proposition 134 tells us that the regulator should set the price in the high season to pH c r and in the low season to pL c Thus efficient pricing requires that highseason consumers pay the marginal operational plus the marginal capacity costs whereas lowseason consumers pay only the marginal operational cost 1335 Peakload pricing over longer periods So far our analysis has concentrated on a time period where there is only one low season and only one high season Suppose that the airline firm is required to invest in capacity for n years n 1 so that capacity holds for n low seasons and n high seasons In this case what would be the profit maximizing pricing structure for this monopoly airline Proposition 135 The monopolys profitmaximizing seasonal pricing and output structure over n low and n high seasons is determined by Thus if the monopoly expects that the capacity would be maintained for n high seasons the effective unit capacity cost in each period should be taken as kn 1336 Limitation of our peakload pricing analysis Some limitations of the traditional approach to peakload pricing analysis are listed in Bailey and White 1974 and Bergstrom and MacKieMason 1991 A serious limitation of this analysis is that we neglected to Page 352 analyze the markets with periodic demand schedules when the different seasonal prices induce consumers to substitute highseason consumption for lowseason consumption High substitutability between peak and offpeak hours is most noticeable in the telephone industry where individuals postpone making personal phone calls until late at night early in the morning and on weekends Thus our analysis is incomplete since it assumes that the demand for peakseason service is independent of the offpeak price 134 Can Firms Control the Seasons Peakload prices are generally calculated by assuming that peak and offpeak periods are exogenously given Although this assumption may describe some public utilities where the regulating authority decides on which periods are considered peak and which offpeak such as electricity and the telephone most firms get to control the quantity demanded in each period by simply adjusting the relative prices in the different periodsseasons For example by substantially reducing winter airfare airline firms can potentially turn a low season into a high season Restaurants control the flow of customers by substantially reducing the price of lunch compared with the price of a dinner Car rental companies can turn the weekend into a highdemand period by substantially reducing weekend rents to attract nonbusinessrelated renters during the weekends All these examples lead to one conclusion namely peak and offpeak periods should be regarded as economic variables and therefore should not be assumed In this section we calculate peakload prices in an environment where the selling firm can use the pricing structure to manipulate which period will be the peak and which will be offpeak We analyze what would be the profitmaximizing pricing structure chosen by a serviceproviding monopoly There are two reasons why we should analyze the monopoly case First analyzing the monopoly case helps us to capture the intuition about the tradeoff between consumers preferences towards certain period services and the cost of maintaining capacity Second many utility and transportation companies are regulated or unregulated monopolies Examples include most transportation companies buses trains and airline PTTs public telegraph and telephone companies and gas and electric utility companies Let us consider an industry selling a particular service in two time periods say during the day denoted by D or during the night denoted by N We denote by pD the price of the service sold during the day and by pN the price of the service sold during the night Page 353 Consumers and seasonal demand Let us consider a continuum of consumers indexed and uniformly distributed on the closed interval a b where and b 1 We denote by δ a particular consumer indexed on a b The utility of consumer δ is assumed to be given by where ß 0 is the reservation utility for a night service Recalling Definition 121 on page 310 we can use the following definition to provide the terminology for characterizing consumers attitudes toward purchasing the service in the different periods seasons Definition 131 Day service and night service arc said to be 1 vertically differentiated if given equal prices PD PN all consumers choose to purchase only the day service 2 horizontally differentiated if given equal prices PD PN consumers indexed by a high δ choose to purchase the day service whereas consumers indexed by a low δ choose to purchase the night service Using 139 we can see that all day and night services are vertically differentiated if since in this case In contrast when the two services are horizontally differentiated according to Definition 131 Finally the consumer indexed by denotes the consumer who is indifferent about whether to buy a day service or a night service at the given market prices for these services Clearly from 139 is determined by Thus given prices all consumers indexed by purchase the night service whereas all the consumers indexed by buy the day service Production of services We denote by nD the number of consumers buying a daytime service and by nN the number of consumers buying a nighttime service Clearly which is the total number of consumers in the economy Page 354 Production of services requires an investment in capacity and in addition bears operation costs For example in transportation industries capacity determines the upper limit on the number of passengers that can be transported in each of the time periods In the telecommunication industry capacity determines the upper limit on the number of phone calls switchboards that can be simultaneously made in each time period Therefore we denote by K the capacity of a serviceproducing firm Then the number of day or night users cannot exceed this capacity that is and We denote by r the cost of a unit capacity facing the firms In addition to capacity cost number of aircraft seats etc serviceproducing firms bear operation costs Therefore we denote by cD the per customer operation cost of producing a day service and by cN the per customer operation cost of producing a night service With no loss of generality we assume that That is the operationpercustomer cost of producing a night service is not higher than the operationpercustomer cost of producing a day service Clearly by varying the relative price of the daytime service and the nighttime service the monopoly serviceproducing firm can shift the peak demand from day to night or night to day For this reason we refrain from using the terminology peak and offpeak periods commonly used in the literature and confine the terminology to daytime or nighttime periods That is peak and offpeak periods are endogenously determined by the selling firm In order to find the profitmaximizing pricing scheme set by the monopoly firm in what follows we decompose the analysis into a cost analysis and a revenue analysis The monopolys cost structure Assuming that all consumers are served either by day or night service we have it that and Then the total cost as a function of the indifferent consumer defined in 1310 is given by Figure 136 illustrates the monopolys production cost as a function of the location of the indifferent consumer Figure 136 shows that the cost is minimized when the market is equally divided between daytime users and nighttime users that is because when the market is equally divided half of the total population buys a day service and the other half buys a night service which implies that the amount of capacity needed by the firm is K a b2 which is at minimum Page 355 Figure 136 Cost structure of a monopoly selling services in two periods Note under this equal division As increases the amount of capacity must increase to accommodate a larger number of nighttime users Hence any deviation from the equal division of consumers either by increasing the number of night users an increase in or by increasing the number of day users a decrease in will result in an additional investment in building capacity If we assume that all consumers are served an increase in means that the monopoly switches consumers from day service to night service Hence for each consumer switching from day to night the monopoly saves an operation cost of cD cN Similarly for each consumer being switched from night to day service a decrease in the operation cost increases by the difference cD cN Altogether in view of 1311 the marginal cost as a function of the indifferent consumer is given by Monopolys revenue The monopoly seeks to extract maximum surplus from consumers Hence in view of 139 the monopoly would charge a price of pN ß for a night service Then according to 1310 determining the price for the day service pD is equivalent to determining the location of the indifferent consumer Hence we can assume that the monopolys choice variable is while pD is determined according to Consequently we can define the monopolys revenue as a function of the location of the Page 356 indifferent consumer by The marginal revenue as a function of the indifferent consumer is given by Figure 137 illustrates the revenue functions for the cases of vertical and horizontal differentiation The bottom figure shows that under vertical differentiation a 1 the revenue is maximized when the indifferent consumer locates to the left of the midconsumer This is because when the products are vertically differentiated all consumers prefer day over night services and given that they are willing to pay more for a daytime service the monopoly will choose prices so that the majority of the consumers will be daytime users Figure 137 Revenue functions for the vertical and horizontal differentiation cases Page 357 The top figure shows that when the products are horizontally differentiated a 1 revenue is maximized when the indifferent consumer locates to the right of the midconsumer The last case when a 1 is not illustrated but in this case the revenue is maximized when the indifferent consumer locates exactly at the midpoint implying that the monopoly allocates half of the consumers to day services and half to night services Monopolys profitmaximizing pricing structure Before we proceed with the calculations of the profitmaximizing pricing structure let us note that the monopolys profit is measured by the distance between the revenue and the cost functions in Figure 137 Figure 137 bottom reveals that under vertical differentiation the monopoly will never choose to price the service so that the indifferent consumer would locate to the right of the midconsumer Figure 137 top reveals that under horizontal differentiation the monopoly will never choose to price the service so that the indifferent consumer would locate to the left of the midconsumer Definition 132 The daytime period is called a peak period if and offpeak otherwise Similarly the nighttime period is called a peak period if and offpeak otherwise Hence Figure 137 and Definition 132 imply that Proposition 136 If the two timeperiod services are vertically differentiated then the monopoly will turn the daytime period into the peak period If the two timeperiod services are horizontally differentiated then the monopoly will turn the nighttime period into the peak period We therefore can state the main proposition concerning monopoly behavior Proposition 137 Given that a monopoly that maximizes profit will set prices so that services are purchased in both periods such that 1 under vertical differentiation 2 under horizontal differentiation Page 358 Proof The monopoly seeks to choose to maximize By Proposition 136 under vertical differentiation Hence 1312 and 1314 imply that Under horizontal differentiation Hence 1312 and 1314 imply that 135 Exercises 1 Congratulations You have been appointed to be the chairperson of the Economics department at Wonderland University Since that old photocopy machine broke down three years ago the department has been deprived of copying services and therefore your first task as a chairperson is to rent copying services from KosKin Xeroxing Services Inc The KosKin company offers you two types of contracts The Department can simply pay 5 cents per page or the department can pay a yearly fee of 300 and in addition pay 2 cents per page a Draw the departments total photocopying expenses as a function of the number of copies made each year under the two types of contracts b Conclude which contract is less costly given the number of copies made each year 2 SouthNorthern Airlines is the sole provider of flights between City A and City B During the winter the inverse demand for flights on this route is given by pw 10 qw where pw is the airfare charged during the winter and qw is the number of passengers flown on this route during the winter Similarly during the summer the inverse demand function is given by ps5qs2 Denote by K the airlines capacity defined by the number of airplane seats SouthNorthern intends to acquire and assume that the average cost of an airplane seat is r 0 Also suppose that the cost of flying each passenger is c 0 a Calculate the number of passengers flown in each season and SouthNortherns profit level assuming that r c 1 b Calculate the number of passengers flown in each season and SouthNortherns profit level assuming that r 3 and c 1 136 References Bailey E and L White 1974 Reversals in Peak and OffPeak Prices Bell Journal of Economics 5 7592 Page 359 Bergstrom T and J MacKieMason 1991 Some Simple Analytics of PeakLoad Pricing Rand Journal of Economics 22 241249 Brown S and D Sibley 1986 The Theory of Public Utility Pricing Cambridge Cambridge University Press Joskow P 1976 Contributions to the Theory of Marginal Cost Pricing Bell Journal of Economics 7 197206 Oi W 1971 A Disneyland Dilemma TwoPart Tariffs for a Mickey Mouse Monopoly Quarterly Journal of Economics 85 7796 Sherman R 1989 The Regulation of Monopoly Cambridge Cambridge University Press Steiner P 1957 PeakLoads and Efficient Pricing Quarterly Journal of Economics 585610 Tirole J 1988 The Theory of Industrial Organization Cambridge Mass MIT Press Page 361 Chapter 14 Marketing Tactics Bundling Upgrading and Dealerships You can automate the production of cars but you cannot automate the production of consumers Walter Reuther In chapter 11 we analyzed advertising as a major marketing tool for firms Chapter 12 analyzed quality durability and warranties as additional strategic tools available to firms Chapter 13 introduced pricing techniques firms use to extract more surplus from the consumers In this chapter we proceed to analyze other important strategic marketing tools available to firms Section 141 Bundling and Tying analyzes the conditions under which a monopoly finds it profitable to sell two or more units of the product bundled in a single package We then proceed to analyze the conditions under which a monopoly finds it profitable to tie the purchase of one product to the purchase of another Next we show that tying can serve another purpose as a tool that a firm employs for the purpose of differentiating itself from competing firms products Section 142 UsedBooks Market analyzes the textbook market and the incentives publishers have for coming out with yearly new editions Section 143 Dealerships analyzes various distribution systems and marketing channels and optimal contracts between producers and distributors Page 362 141 Bundling and Tying Bundling refers to a marketing method in which firms offer for sale packages containing more than one unit of the product Thus a firm is said to bundle if consumers have to choose between buying a number of units of the product at a given price or not buying at all We sometimes say that a firm that bundles is engaged in nonlinear pricing meaning that each unit of the product is not sold for the same price Examples of bundling include all quantity discounts buy one unit and get the second one for free volume discounts on phone calls and frequentflyer mileage earned by passengers who convert them to free tickets Tying refers to firms that offer for sale packages containing at least two different products For example a car dealer may offer cars with an already installed car radio and a computer dealer may include some software packages with the sale of computer hardware In this case we say that the seller ties complementary products However not every instance of tying involves complementary products for example a book store may provide a Tshirt to a customer who purchases a book 1411 How can bundling be profitable Consider a monopoly selling a product to a single consumer whose demand curve is given by Qp 4 p where p is the monopolys price and Q is the quantity purchased Assuming that production is costless we showed in chapter 5 that the monopoly will set pm 2 and sell Qm 2 yielding a profit level of πm 2 2 4 Figure 141 illustrates the monopoly profitmaximizing price and quantity sold Clearly since this monopoly cannot price discriminate with respect to quantities the consumer surplus is positive and is equal to 2 Figure 141 Bundling monopoly Page 363 Recall from subsection 323 on page 52 that consumer surplus is defined by the area of the triangle above pm in Figure 141 Now suppose that the monopoly bundles four units of the product in a single package and offers it for sale for 8 minus 1 cent per package of four units The consumers problem now is whether to purchase the package for 8 minus 1 cent or not to purchase at all Since purchasing the package of four yields a consumer surplus of 4 42 8 it is clear that a consumer faced with this decision would prefer to purchase the package of four rather than not purchasing at all Therefore in the case of bundling the monopolists profit is 8 4 πm Hence Proposition 141 A monopoly engaging in bundling would extract all consumer surplus and will therefore make a higher profit than a nonbundling monopolist Therefore a bundling monopolist earns the same profit as a perfectly discriminating monopoly 1412 How can tying be profitable We now show that if consumers are heterogeneous in the sense that they have different valuations for different products firms can increase their profits by selling the different products in one package The gain in profit from tying is analyzed in Burstein 1960 Adams and Yellen 1976 Lewbel 1985 and McAfee McMillan and Whinston 1989 We analyze the gains from tying by examining the following monopoly example Consider a monopoly selling two goods labeled X and Y There are two consumers denoted by i i 1 2 who buy at most one unit of each product and have different valuations for the different products We denote by the consumer is valuation of product X the maximum price consumer i is willing to pay for product X Similarly denotes consumer is valuation of product Y Table 141 shows the different valuations of the two consumers Table 141 shows that consumer 1 is good Xoriented whereas consumer 2 is good Yoriented however both consumers gain utility from both goods Product X Y Consumer 1 Consumer 2 Table 141 Consumers valuations for tied products H L 0 Page 364 Finally we assume that the consumers purchase the products only for consumption and therefore do not trade with each other No tying When tying is not allowed the monopoly has two options First it can set a low price and sell both products to both consumers Second it can set a high price for both goods and sell one unit to each consumer Formally suppose that the monopoly sets PX pY L Then both consumers purchase both goods yielding a profit of πNTL 4L Now suppose that the monopoly sets pX pY H In this case consumer 1 buys only good X and consumer 2 buys only good Y Hence πNT 2H Comparing the two profit levels yields the monopoly price decision for the case of no tying Thus when H is high a monopoly would increase the price to a high level and sell only two units When H is close to L the monopoly would reduce the price and sell four units Tying Now suppose that the monopoly decides to sell only packages that contain one unit of good X and one unit of good Y for a price of pT Clearly the monopoly sets pT H L in order to extract all surplus from the consumers In this case the monopoly sells two packages and earns a profit of πT 2H L Therefore we can state the following proposition Proposition 142 A monopoly selling two products to heterogeneous consumers whose preferences are negatively correlated makes a higher profit from selling a tied package than from selling the components separately Formally for every H L 0 πT πNT Proposition 142 demonstrates that bundling enables a monopolist to earn the profit level of a price discriminating monopolist as long as the preferences of consumers are negatively correlated In addition the gains from tying increase when the preferences become more diverse H L increases 1413 Mixed tying Adams and Yellen 1976 demonstrated that a tying monopoly can further increase its profit if in addition to pure tying the monopoly sells Page 365 only packages composed of both products the monopoly sells the two products separately Following the example of Table 141 let us consider the threeconsumer example given in Table 142 We now investigate the monopolys marketing options under no tying NT pure tying T and mixed tying MT Product X Y Consumer 1 Consumer 2 Consumer 3 Table 142 Consumers valuations for the mixedtying example No tying There are two possibilities under no tying 1 If pX pY 3 then consumer 1 buys good X consumer 3 buys good Y and consumer 2 buys one unit of X and one unit of Y Therefore the monopoly profit is φNT 3 4 12 2 If pX pY 4 then consumer 1 buys good X consumer 3 buys good Y and consumer 2 does not purchase any good Therefore the monopoly profit is φNT 4 2 8 It is clear that option 1 will be chosen by the monopolist since this option yields a profit of πNT 12 Pure tiling In the case of pure tying the monopoly sells packages that contain one unit of X with one unit of Y for a single price denoted by pT Again there are two possibilities 1 The monopoly sets pT 4 In this case all three consumers purchase the tied package Hence πT 3 4 12 2 The monopoly sets pT 6 This price exceeds the package valuations of consumers 1 and 3 and only consumer 2 buys this package Hence π 1 6 6 It is clear that the monopoly will choose option 1 since it yields a profit level of πT 12 Page 366 Mixed tying We now assume that the monopoly markets the two products in two forms It sells a package of one unit of X and one unit of Y for the price of pMT 6 in addition the monopoly offers the individual products for sale thus mixed tying Suppose that the monopoly sets the price of the individual products to pX pY 4 Clearly consumer 2 will not buy the individual products since each product is priced at 4 which is above his valuation level However consumer 2 will purchase the tied package for pMT 6 In addition consumer 1 will purchase good X and consumer 2 will purchase good Y each priced at 4 Thus the total monopoly profit under mixed tying is given by πMT 1 6 4 4 14 12 πT Hence Proposition 143 Mired tying may yield strictly higher profit levels than pure tying and no tying marketing tactics The intuition behind Proposition 143 is as follows Consumer 2 can be viewed as a consumer who treats the two products as complements since he attaches a relatively low valuation to each product but values the two products together at 6 which exceeds the valuation of a package by consumers 1 and 3 In contrast consumers 1 and 3 do not attach any extra value to the package but attach a high value to one of the products By using mixed tying the monopoly can extract maximum surplus from consumer 2 by selling him his desired package and can extract all surplus from consumers 1 and 3 by selling them their most desired product Finally as pointed out in Adams and Yellen 1976 it is possible to show that mixed tying is not always as profitable as pure tying 1414 Tying and foreclosure As we will discuss in section 144 antitrust laws prohibit bundling or tying behavior whenever it leads to a reduced competition in the industry Following Seidmann 1991 and Whinston 1990 we ask why does antitrust law assume that bundling and tying may reduce competition That is what is the connection between tying and reduced competition To see this connection we look at two consumers type 1 and type 2 and a twosystem example Suppose that consumers desire to purchase a system that combines one unit of a computer hardware and one monitor There are two firms producing computers which we denote as brand X and brand Y and one firm producing monitors denoted by Z We assume that monitors are compatible with both brands X and Y see section 103 for an analysis of components compatibility Page 367 The consumers preferences are given by Thus 142 assumes that consumer 1 is brand Xoriented whereas consumer 2 is brand Yoriented Three independent firms Suppose now that the firms producing brands X Y and Z are independently owned We look for a NashBertrand equilibrium Unfortunately there is more than one equilibrium corresponding to a high monitor price and low computer prices or a low monitor price and high computer prices Therefore Proposition 144 When the industry is decomposed into three independent firms 1 The following prices constitute a NashBertrand equilibrium pX pY 2 pZ 1 In this equilibrium the firm producing X sells one unit to consumer 1 the firm producing Y sells one unit to consumer 2 and the firm producing Z sells two units one unit to each consumer the firms earn profit levels of πX πY πZ 2 2 The above equilibrium is not unique Proof If firm Z raises its price no consumer would buy any system Also since all consumers already buy a unit of Z firm Z cannot increase its profit by lowering its price In order for the X producing firm to undercut the Yproducing firm it must set px 0 and hence cannot increase its profit Therefore undercutting is not profitable to firms X and Y This establishes part 1 To establish part 2 observe that the following triplets are also equilibria pX pY pZ 1 1 2 pX pY pZ 0 0 3 and pX pY pZ 3 3 0 Firm X takes over firm Z We now show that firm X can drive firm Y out of business when it buys firm Z and sells products X and Z tied in a single package Suppose that Page 368 the newly merged firm denoted by XZ offers the package containing one unit of X with one unit of Z for a price of pXZ We now state our main proposition Proposition 145 1 By setting the package price to Pxz 3 the firm selling the package XZ drives firm Y out of business Thus tying can serve as a tool for foreclosing a competing firm 2 Foreclosing is not profitable for the bundling firm The profit of the merged firm XZ when engaged in foreclosing firm Y is lower than the sum of the two premerged firms X and Z Proof Suppose that firm Y sets pY 0 When pXZ 3 the utility for consumer 2 when buying system XZ and product Y for pY 0 is U2 3 pXZ 0 0 Hence firm Y will not produce and consumer 2 is not served This proves part 1 Under this foreclosure equilibrium πXZ 3 However the sum of the profits of firms X and Z before the merger was larger than 3 Proposition 145 shows that tying for the purpose of foreclosing a horizontally competing firm is too costly to the foreclosing firm and is therefore unlikely to be used The proposition also showed that the act of foreclosing the market reduces aggregate industry profit since the foreclosure causes one consumer not to be served Thus a foreclosed industry may be serving a reduced number of consumers and hence earns a lower aggregate industry profit However in what follows we show that when firm X buys firm Z then it is profitable for the merged firm not to completely foreclose on the competing firm but to leave it a small market share We therefore define the concept of foreclosure Definition 141 Suppose that firm X buys firm Z Then firm X is said to be foreclosing firm Y if for any given small there exists a Nash equilibrium in prices pXZ and pY that would leave firm Y with a profit of Definition 141 states that foreclosure implies that firm Y can still profitably sell units of product Y However the merged XZ firm could set pXZ so that it can bring the profit of firm Y to as low as it wishes Proposition 146 1 Let be a small number The prices and constitute an foreclosure equilibrium Page 369 2 An foreclosure equilibrium yields a higher profit level to the foreclosing firm than does the total foreclosure equilibrium given in Proposition 145 Proof Clearly these prices constitute a Nash equilibrium To demonstrate the profit advantage of this equilibrium over the total foreclosure equilibrium let us observe that firm XZ sells to both consumers and therefore earns However under the totalforeclosure equilibrium firm XZ sells to only one consumer thereby earning πXZ 3 Thus for a sufficiently small the foreclosure equilibrium is more profitable for the foreclosing firm The intuition behind the profitability of the foreclosure equilibrium is that the foreclosing firm manages to provide the Yoriented consumer his or her most preferred system That is under the foreclosure equilibrium consumer 2 buys system XZ for and then discards the X component and buys the Y component for a negligible price However under the total foreclosure equilibrium described in Proposition 145 consumer 2 does not get his most preferred system and therefore since his willingness to pay falls to 1 consumer 2 does not buy any product 1415 Tying and international markets segmentation Firms selling in different markets will generally find it profitable to price discriminate among the markets in which they sell The profitability from price discrimination has already been discussed in section 53 The problem is that in order to price discriminate the markets should be segmented in the sense that consumers or merchants should not be able to buy the product in the lowprice market and then sell it in the highprice market That is in order for price discrimination to be feasible agents should be prevented from engaging in arbitrage activities In the international economy arbitrage is weakened by heavy trade restrictions imposed by all governments Restriction methods include tariffs quotas valueadded taxes foreign exchange dollar holdings restrictions safety regulations and the usual bureaucracy These government restrictions help firms to engage in price discrimination across international boundaries The question that we ask in this section is whether the removal of trade barriers such as that practiced in the European Community and the North America would imply that the prices of products would necessarily equalize across markets We show that since market integration such as that in the EC cannot remove all differences of language culture and location among consumers firms may have at their disposal Page 370 means for making international arbitrage costly to consumers even after integration Whenever the regional markets differ in some aspects firms may find it possible and profitable to segment the markets themselves to exploit these differences especially after government trade restrictions are removed Consider a twocountry world economy with one consumer in each country There is only one product produced and distributed by a single manufacturer This worldmonopoly producer has two marketing options It can sell directly to the consumer in each country or it can open a dealership in each country selling the product tied with service to the consumer Let and denote the prices of the product when tied or not tied with services respectively The utility function of the consumer in country k k 1 2 is given by where Bk measures the maximum amount a consumer in country k is willing to pay forthe basic without service product Thus each consumer treats a product tied with domestic service and the product with no service as vertically differentiated see section 122 for a definition since for equal prices each consumer would purchase only the serviced product The following assumption would make price discrimination profitable for the international monopoly Assumption 141 The consumer in country 1 is willing to pay a higher price for the basic product than the consumer in country 2 Formally B1 B2 Finally with no loss of generality we assume that the product itself is costless to produce but that there may be costs associated with each dealership providing services in each country which we denote by w In the next two subsections we compare two marketing strategies available to the international monopoly No attempts to segment the market Suppose that the monopoly sells the product directly to each consumer say via international mail order and hence does not tie any local service with the product Then given zero transportation costs there is perfect arbitrage between the countries and therefore the monopoly Page 371 would charge identical prices in both countries Otherwise a consumer living in the lowprice country would make a profit by buying the product in his country and selling it in the highprice country Altogether the international monopolys world uniform price and profit levels under no segmentation NS are given by Thus if consumers valuations are not too diverse B1 2B2 then the monopoly would find it profitable to lower the price to B2 and sell two units If the consumers valuations are substantially diverse B1 2B2 then the international monopoly would raise the price and sell the product only in country 1 Segmenting the market We now suppose that the international monopoly opens dealerships in each country selling the product tied with local service That is the local dealer may produce manuals provide training using the local language and provide a repair service for the product Since local services are not internationally traded the monopoly can charge a different price in each country Thus under segmentation S the price in each country k k 1 2 and the international monopoly profit level are given by Thus by tying services with the product the international monopoly is able to segment the markets and hence to price discriminate between the markets Comparing the profit levels given in 145 and 144 yields Proposition 147 A sufficient condition for the international monopoly to segment the international market into two national markets bit providing local services is that B1 B2 2w σ Is there room for arbitrage after segmentation The prices given in 145 differ by country That is However in order to prove that this segmentation is sustainable we still need to prove that under these prices arbitrage will not occur In other words following Horn and Shy 1995 we need to show that the highvaluation consumer consumer 1 will not benefit from traveling to country 2 buying the product for B2σ then taking it to country 1 and consuming it without the service part Now the consumer in the highprice country consumer 1 will not benefit from purchasing the product with service in country 2 and then Page 372 using it without service in country 1 if that is if the utility from doing that is lower than the utility of buying from the local dealer with service at a high price Substituting 145 into this condition yields that is if the difference in basic valuation of the product between the countries does not exceed the value of local service Equation 146 yields our major proposition Proposition 148 If consumers valuation of service is higher than the differences between the two consumers in the basic product valuation ie B1 B2 σ then the international monopoly will succeed in segmenting the international market into two national markets in the sense that equilibrium price differentials between the two national markets will not generate arbitrage activities 1416 Tying as product differentiation So far we have analyzed how a monopoly can increase its profit by using tying and mixed tying In this subsection we analyze how tying tactics are used by firms competing in prices in a market for a homogeneous product We show that under oligopoly firms may use tying tactics in order to differentiate themselves from the competing brands Put another way we show that firms engaging in a Bertrand competition in homogeneous products can increase their profit by tying their product with another product or a service in order to differentiate itself from the competing firm This strategy may lead to market segmentation where the market is split into a group of consumers buying the homogeneous product and another group buying a product tied to a service contract Following Carbajo de Meza and Seidmann 1990 and Horn and Shy 1996 we consider two firms that produce identical products However the firms can sell the product with or without service By service we mean service repair contracts warranties help in learning how to operate the product and so on Examples of firms that sell products without service include some mailorder firms that sell products via the mail without providing substantial training or assembling services to the customers Let us consider a single market for a homogeneous product sold by two firms The firms have the option of selling the product with or Page 373 without supporting services Let pS denote the products price when tied with services and pN the price when sold untied Consumers attach the same value B to the basic product Services however yield different benefit to different consumers To capture this variable let consumers be uniformly distributed with a unit density on the unit interval according to an increasing valuation for services A consumer indexed by s 0 derives the least benefit from services whereas the consumer indexed by s 1 derives the most Each consumer buys at most one unit of the product and we assume that B is large enough relative to consumers reservation utilities so that in equilibrium everyone buys a unit The utility of consumer s is given by Thus the servicetied product is vertically differentiated from the basic product in the sense that if both are sold for the same price each consumer prefers to have the service bundled with the product see Definition 121 on page 310 for a definition of vertical differentiation Let m 0 denote the unit production cost of the basic product and let w 0 denote the production cost of services influenced say by the wage rate in the services sector For the rest of this analysis assume that w 2 as we will see this is a necessary and sufficient condition to guarantee that each firm will have a nonzero market share The interaction between the firms takes place in two stages it is a twostage game First each firm decides whether to sell the product with or without a unit of services In the second stage the firms compete in prices Solving for a subgame perfect equilibrium Definition 210 on page 27 we first characterize the secondstage pricecompetition equilibrium under three types of outcomes arising in the first stage of the game Both firms tie services or neither does Suppose now that neither firm ties its product with services Since the products are homogeneous they are sold at a uniform price of pN m both firms make zero profits and the market can be arbitrarily divided between the firms If both firms tie the product with services the products become homogeneous again and will be sold at a price pS m w Hence both firms make zero profits and the market can be arbitrarily divided between the two firms Page 374 One firm ties services If one firm sells the good tied with services and the other without services and if each firm sells a positive amount then the marketdividing condition is given by where is the market size and share of the nonserviced product whereas is the market size and share of the firm that ties Hence Thus a firm that tiesin has a profit and a firm that does not tiein has a profit where is defined by 148 We define an equilibrium in the second stage when one firm ties and the other does not as the pair such that for a given the bundling firm chooses to maximize subject to satisfying 148 and for a given the nontying firm chooses to maximize subject to satisfying 148 Substituting from 148 into the profit functions and then maximizing with respect to corresponding prices yield firstorder conditions for the interior solution given by Therefore the reaction functions are given by respectively Solving the middle parts of the reaction functions given in 1410 shows that an interior solution exists and is given by Page 375 The first stage Tying versus not tying Equations 1411 imply that when one firm ties with services and the other does not both firms make positive profits in contrast to the case where both bundle or both do not Hence Proposition 149 1 In a twostage game where firms choose in the first period whether to tie their product with services one firm will tiein services while the other will sell the product with no service 2 An increase in the wage rate in the services sector would a increase the market share of the nontying firm the firm that sells the product without service and decrease the market share of the tying firm decreases b increase the price of the untied good and the price of the tied product both and increase Part 2a of Proposition 149 is intuitively clear An increase in the wage rate the cost of providing services would reduce the market share of the firm that ties service with the product Part 2b is interesting since it implies that an increase in the wage rate would raise all prices including the price of the firm that does not provide services the firm that does not pay the w This happens because prices are strategic complements see Definition 72 meaning that when the cost and therefore the price of the tying firm increases the price of the nontying firm also increases The socially optimal provision of service We now turn to ask whether from a social point of view this duopoly equilibrium results in too much or too little service marketed to consumers That is the interesting question is whether the amount nonserviced products is too high or too low from a social welfare perspective The socially optimal number of consumers purchasing the product without service denoted by s is obtained under marginalcost pricing Thus let pS m w and pN m Then It can easily be verified that if and only if Hence Proposition 1410 1 If the wage rate in the services sector is high that is when the equilibrium number of consumers purchasing the product tied with service exceeds the socially optimal level That is Page 376 2 If the wage rate is low that is when the equilibrium number of consumers purchasing the product tied with service is lower than the socially optimal level That is Proposition 1410 is easy to interpret When the cost of service production w is high a smaller number of the serviced product is socially desirable that is the firm that ties the product with service overproduces from a social view point This is interesting since under a high wage rate one would expect the sales of the servicetying firm to overtaken by the discount firm that sells with no service However as we show below the nonservicing firm takes an advantage of the servicing firms high serviceproduction cost and raises its price thereby losing market share to the highcost servicing firm To support the last argument let us investigate which firm charges a higher markup the service tying firm or the discount nonservicing firm We define a firms price markup by the ratio of selling price minus the unit production cost divided by the unit production cost Hence for w 12 Therefore Proposition 1411 When w 12 and when one firm ties its product with services while the competing firm sells an identical product without services the firm that sells without service say the discount or mail order firm charges a higher markup Proposition 1411 provides the key intuition behind this price competition since it shows that it is the discount or mailorder firm that overcharges relative to cost In other words it demonstrates that the discount stores use the fact that they have a monopoly on those consumers who do not desire services low s consumers Thus the nonservicing firm pushes the price up to the point where the price of a nonserviced product is close to the price of a serviceinclusive product 142 Killing Off Markets for Used Textbooks Perhaps the most challenging marketing task is to market a new product in a market flooded with perfectly functioning used products The reason for this is that marketing often relies on advertising the sustained quality and durability of the product hence if consumers believe that the advertising is reliable then consumers will be convinced that old Page 377 products need not be replaced Thus advertising the quality of the product may be counterproductive for a firm trying to sell new products Since markets for used products often kill off the market for new products manufacturers are forced into special marketing techniques to convince consumers to drop their used products and replace them with new ones A notable example of this process is the market for light private aircraft Light aircraft happen to be extremely durable because the airframe rarely degrades and engines are routinely replaced This reduction in demand caused several aircraft makers to stop the production of small aircraft eg of the twoseater Cessna 152 It is often thought that textbook publishers come up with yearly revisions in order to prevent the usedbooks market from taking sales away from the publishers Benjamin and Kormendi 1974 Liebowitz 1982 Miller 1974 and Rust 1986 have all analyzed the market for used and new textbooks We investigate this problem by considering a simple twoperiod model The students Suppose that in each period t t 1 2 there are n students who are requested by their professor to purchase a textbook for their class which will conclude at the end of the same year That is in period 1 there are n students who purchase a new textbook the students graduate at the end of period 1 and offer the books for sale to the n period 2 newly entering students We assume that the value of new and used book to an entering student is V We postpone to an exercise in section 146 analyzing the case where used books are less valuable to students than new ones We denote by pt the period t price of a book t 1 2 Thus the utility of a generation t student is given by We assume that students have perfect foresight meaning that they are able to calculate the profitable actions taken by the book publisher in the second period The textbook monopoly We assume that there is only one textbook publisher for this particular course and that in period 1 the publisher sells a brandnew textbook We denote the unit production cost of a book by c c 0 In addition in the second period the monopoly can invest an amount of F to revise Page 378 the textbook and to introduce a new edition that may be required by the professor Altogether in period 1 the monopoly chooses the price for the new book p1 and in period 2 the monopoly chooses whether or not to introduce a new edition and the corresponding price or 1421 Secondperiod actions taken by the textbook publisher Suppose that all the n period 1 students have purchased a textbook in period 1 and that they offer them for sale as used books in period 2 The monopoly seller has to decide whether to invest F in order to introduce a new edition in this case the value of a used textbook drops to zero or to sell new copies of the old edition Introduction of a new edition If a new edition is introduced and adopted by the professor the value of used books drops to zero so none of the n period 1 students are able to sell their used books Hence all the n period 2 students purchase new books for the monopoly price In this case the secondperiod profit of the monopoly publisher is given by Selling the old edition When a new edition is not introduced the publisher and the n period 1 students compete in homogeneous products However given that the n period 1 students already own the used books their production cost is zero compared with a unit production cost of c 0 for the monopoly Hence Bertrand price competition see section 63 drives the usedbooks price to unit cost Formally Thus the fact that period 1 students do not desire their used textbooks enables period 1 students to undercut the publisher and to sell the used books to period 2 students Note that the assumption that the number of students does not vary between generations is critical 1422 Profit of the publisher We now calculate the monopolys sum of profits both for when a new edition is introduced in the second period and for when it is not Clearly from 1414 and 1415 we know that a new edition is introduced in Page 379 period 2 if Hence if condition 1416 is not satisfied the monopoly calculates that the textbook sold in the first period will be sold as used in period 2 for a price of In this case the monopoly charges P1 V c which is the value of the book plus the resale value in period 2 If condition 1416 holds then a new edition will be introduced in period 2 so the firstperiod monopoly price is only p1 V since textbooks will not have a resale value in period 2 Altogether the sum of the twoperiod profits is Equation 1417 shows the profit of the publisher under the two possible outcomes a new edition is introduced or it is not introduced in period 2 1423 Welfare in the textbook market We now wish to compare the welfare under the two outcomes Table 143 shows the utility of each generation of students under the two textbooks outcomes Table 143 shows that the publisher absorbs all consumer Publishers Action Generation ts Utility tl t2 2 0 0 nV V c c nV c Table 143 Consumers utility under the new and used textbooks outcomes surplus when he introduces a new edition However when the Publisher does not revise the book period 2 students gain a strictly positive surplus since competition with used books reduces the price to cost p2 c We define the economys social welfare as the sum of utility and profit levels over the two periods Summing up Table 143 and 1417 yields Thus Proposition 1412 A new edition is socially undesirable Page 380 The result given in Proposition 1412 is not surprising since in our model new editions do not serve any social purpose However given that the usedbooks market introduces competition to publishers the publisher introduces new editions in order to disconnect from the usedbooks market 143 Dealerships Manufacturers are often not involved with direct marketing to consumers generally referred to as end users Instead manufacturers sell their products to dealers and distributors who offer the products for sale at retail prices In the literature the types of arrangements between manufacturers and retailers are referred to as vertical restraints and are surveyed in Mathewson and Winter 1986 and Tirole 1988 chap 4 The common arrangements between manufactures and distributors are 1 exclusive territorial arrangements where a dealer is assigned a territory of consumers from which other dealers selling the manufacturers product are excluded 2 exclusive dealership which prohibits the dealer from selling competing brands 3 fullline forcing where the dealer is committed to sell all the varieties of the manufacturers products rather than a limited selection and 4 resale price maintenance where the dealer agrees to sell in a certain price range which is generally a minimum or a maximum price required by the manufacturer All these arrangements are accompanied by payment arrangements specifying how the dealers pay the manufacturer for the product they sell such as a special dealers price that the dealer pays the manufacturer for each unit it sells or acquires for stocking a franchise fee or a lumpsum fee that the dealer pays the manufacturer irrespective of the number of units the dealer sells or joint ownership under which the manufacturer partially invests in establishing the dealership maintains part of the ownership and therefore receives a share of the profit according to the manufacturers ownership share In this section we will not investigate the reasons why manufacturers do not engage in direct marketing Reasons for this behavior include increasing returns in distribution due to consumers shopping needs such as choice of variety and needs for services integration of various complementary products produced by different manufacturers into systems usable by consumers and geographical locations Therefore in what follows we assume that marketing through dealers is profitable to the producers and concentrate on the various contracting arrangements among producers and distributors Page 381 1431 Dealership distributing at a single location Consider a market for a homogeneous product The demand for the product is linear and is given by p a Q or Q a p where p denotes the price and Q the quantity purchased On the supply side we assume a manufacturer who sells a homogeneous product to a single distributor who is the sole seller of the product In what follows we examine various contracts between the manufacturer and the dealer Doublemonopoly markup We start with a simplest contract under which the manufacturer sells each unit to the dealer for a price of d dollars Let us assuming that the dealer has no other costs the dealer treats d as his or her unit cost of production Being an exclusive dealer for the product the dealer acts as if he or she were a monopoly with a unit production cost of d Thus the dealer chooses the number of units he or she sells that solves the monopoly problem given by The firstorder condition is given by Hence the number of units sold the consumer price and the profit of the dealer are given by The purpose of this analysis is to investigate what the manufacturers profitmaximizing per unit price d charged to the dealer should be With a unit production cost of c the manufacturers profit level is whereQd is a function of d and is determined by the dealer according to 1420 Hence the problem of the manufacturer is to choose d that solves The firstorder condition is given by Page 382 Substituting 1422 into 1420 and then into 1421 yields the number of units sold by the dealer and the profit levels of the dealer and the manufacturer Hence We now state our main proposition Proposition 1413 When a monopoly manufacturer sets a per unit price to be collected from the dealer for each unit sold then 1 the manufacturer earns a higher profit than the dealer 2 the manufacturer could earn a higher profit if he does the selling by himself Moreover the total industry profit the manufacturers plus the dealers is lower than the profit earned by a single manufacturerseller monopoly firm Proof Part I is given in 1423 To prove part 2 recall from section 51 that a monopoly that produces and sells deals its product earns a profit of In other words the profit of a directmarketing monopoly exceeds the sum of manufacturer and dealers profits when marketing is done via a dealership The reason for this difference is that under a dealership there are two markups one markup set by the manufacturer and a second markup by the dealer These markups raise the enduser price above the pure monopoly price and reduce the quantity sold below the pure monopoly level Twopart tariff contracts So far we have seen that establishing a dealership reduces the aggregate industrys profit and in particular the profit of the manufacturer More precisely the manufacturer who sells his products through independent dealerships is concerned with two major issues How to induce the dealer to choose a relatively low price and How to extract shift the profit from the dealer We now show that using a twopart tariff see section 131 contract between the manufacturer and the dealer can result in no loss of profit to the manufacturer The problem of the manufacturer is to offer a contact that will be acceptable to the dealer but will induce the dealer to charge the pure Page 383 monopoly price We show now that a contract in which the manufacturer sells each unit of output to the dealer for d c unit production cost but in which the dealer has to pay in addition a lump sum participation fee denoted by φ may result in a monopoly profit to the manufacturer and no loss to the dealer Proposition 1414 A twopart tariff contract with yields the pure monopoly profit to the manufacturer and no loss to the dealer Proof Under this contract the dealer maximizes yielding a firstorder condition given by 0 a 2Q d which under d c yields the pure monopolys output level and a revenue level of a c24 Hence πd 0 and all the monopoly revenue is paid to the manufacturer as a lump sum fee φ 1432 Resale price maintenance and advertising In general resale price maintenance is an agreement between the dealers and the manufacturer to maintain a price floor minimum price a price ceiling maximum price or a fixed enduser price From the manufacturers point of view resale price maintenance has two goals First it can partially solve the low industry profit associated with the manufacturer and dealers double markup as demonstrated in Proposition 1413 Second it can induce the dealers to allocate resources for promoting the product In this subsection we demonstrate another type of possible arrangement between the manufacturer and two potentially competing dealers Let us consider a market for a product where the demand is affected by the industry aggregate advertising level which we denote by A Forreally assume that the demand for the product is given by where p denotes the market price and Q the aggregate quantity sold Suppose now that the manufacturer sells the product to two dealers competing in prices As before we denote by d the per unit price at which the manufacturer sells to dealers Also denote by Ai the expenditure on advertising by dealer i i 12 Hence the aggregate advertising spending level is given by A A1 A2 Our benchmark equilibrium is stated in the following proposition Proposition 1415 Suppose that the manufacturer is not engaged in advertising and suppose that the manufacturer sells each unit of the Page 384 product to the two dealers for the price of d per unit Then for any given d no dealer would engage in advertising Ai 0 i 12 and the demand would shrink to zero so no sales are made Proof Since the two dealers are engaged in a Bertrand price game for homogeneous products see section 63 the price would drop to a unit dealers cost Therefore p d Hence for every given value of d each dealer makes zero profits even without spending on advertising Consequently dealers will not advertise We now show that a type of arrangement called resale price maintenance can eliminate price competition among dealers and induce them to engage in advertising In the present case suppose that the manufacturer mandates a price floor to both dealers that we denote by pf Clearly the manufacturer must set since otherwise dealers would make negative profits even without engaging in advertising Given the price pf the quantity demanded is given by which is assumed to be equally split between the two dealers That is advertising in this model is assumed to raise directly the demand faced by the industry only indirectly the demand faced by the individual dealer The only strategic variable of each dealer is the advertising level Formally each dealer i takes the advertising level of the competing dealer Aj as given and chooses his or her advertising level that solves The firstorder condition with respect to Ai yields implying that Equation 1425 shows that only the aggregate amount of advertising can be determined for given levels of pf and d and the distribution of advertising spending among the dealers is not uniquely determined in the sense that for every extra dollar dealer 1 spends on advertising dealer 2 reduces the amount spent on advertising by exactly one dollar However 1425 implies our main proposition Proposition 1416 Resale price maintenance pf d ensures that at least one dealer will engage in advertising Moreover the aggregate dealers spending on advertising increases with an increase in the gap between the price floor and the dealers per unit fee pf d Page 385 1433 Territorial dealerships We now investigate how territorial considerations affect a manufacturers derision whether to grant a single dealership or whether to grant dealerships to several dealers who may engage in competition over a given territory We assume that the manufacturers production cost is zero c 0 and that the manufacturer sells each unit of the product to each dealer for a price of d to be determined by the manufacturer In addition each dealer has to invest an amount of F 0 in order to establish a dealership Consider a city with two consumers located at the edges of town as illustrated in Figure 142 We assume that the transportation cost from Figure 142 Territorial dealerships in the linear city Up A single dealership locating at the center Down Two dealers locating at the edges of town an edge of town to the center is measured by T Hence the transportation cost from one side of town to the polar side is 2T Let B B F T denote the basic value each consumer attaches to the product We assume that the utility function of each consumer i i 1 2 is given by Exclusive territorial dealership located at the town center The dealer buys each unit of the product from the manufacturer at the price of d and chooses the price pD to maximize profit Being a Page 386 monopoly over the entire town the monopoly extracts all consumer surpluses by charging pD B T which by 1426 is the maximum price a consumer is willing to pay when shopping at the center of town Hence the dealer sells to the two consumers QD 2 and earns a profit of πD 2 pD d F 2B T d F The manufacturers problem is to set the dealers per unit fee to maximize profit subject to having the dealer making a nonnegative profit Formally the manufacturer solves Two dealerships located at the town edges Equilibrium of a price game Suppose now that the manufacturer grants dealerships to two stores located at the edges to town Our analysis will focus on two sizes of towns reflected in the transportation cost parameter T Definition 142 The town is said to be large if T F4 and small if T F4 In Proposition 78 on page 160 we proved that a NashBertrand equilibrium prices may not exist for the discretelocation model Therefore we look for an equilibrium concept where equilibrium prices satisfy the condition that no dealer would find it profitable to lower the price to undercut a rival dealer selling at the other side of town If we recall our definition of an undercutproof equilibrium given in Definition 75 on page 161 then Definition 143 The pair of prices and is called an undercutproof equilibrium UPE if That is each dealer selling to the consumer nearby does not find it profitable to undercut the rival dealer by selling at the rivals price minus the transportation cost of crossing the whole town Two dealerships Large town case When the town is large F 4T we show that firms cannot increase their profits by engaging in undercutting simply because subsidizing the Page 387 transportation cost of the consumer located on the other side of town is too costly Therefore the manufacturer can extract maximal rent by setting the dealers fee to d B F Hence each dealer charges the maximal price and earns and the manufacturer earns We need to ensure that the prices set by the dealers constitute a UPE Definition 143 This is easily established by observing that which holds if F 4T which is implied by our assumption that the town is large Comparing the manufacturers profit level with the profit given in 1427 implies Proposition 1417 When the town is large the manufacturer will grant a single dealership to be located at the center if 2T F 4T and two dealerships to be located at the edges of town if F 2T Proposition 1417 simply states that if the sunk cost associated with establishing a dealership is high the manufacture will establish only one dealership Two dealerships Small town case When the town is small F 4T the two dealerships are engaged in an intense price competition which yields losses to the two dealers To see this solving the two UPE conditions in Definition 143 yields that Therefore since the town is small Hence the dealers and the manufacturer cannot make positive profits Proposition 1418 When the town is small the manufacturer will grant only a single dealership to be located at the center Imposed territorialexclusive dealerships The previous analysis showed that when the town is small the manufacturer cannot make any profit when he or she grants dealerships to two dealers Therefore we now ask what kind of arrangements can be made between the manufacturer and the dealers so that the two dealers could locate at the edges of town but would refrain from price competition leading to a reduced industry profit Clearly if the dealers could collude in prices they could charge the local monopoly price and absorb all consumer surpluses However if they cannot collude then we ask what kind of contracts the manufacturer can write with the dealers that would ensure that dealers charge the local monopoly price One way of doing that is given in the following proposition Page 388 Proposition 1419 Suppose that the manufacturer grants dealerships to two dealers located at the edges of town Then granting territorialexclusive dealerships exclusive dealerships limited to geographical locations yields a strictly positive profit to the manufacturer Proposition 1419 does not require a formal proof because if the manufacturer limits the territory of dealer 1 to selling only on 012 and of dealer 2 to selling on 121 each dealer becomes a local monopoly and charges i 1 2 The manufacturers problem under territorialexclusive dealership is to set the unit price d it sells to dealers that solves implying that a per unit fee of dM B F hence πM 2B F 0 Note that the same profit levels could be achieved by simply using a resalepricemaintenance mechanism RPM analyzed earlier in subsection 1432 In other words the manufacturer could set a consumer price floor of thereby preventing the dealers from engaging in price competition Finally note that although territorialexclusive dealerships increases profit over the competiting dealerships case in a small town the manufacturer can make a higher profit by simply granting a single dealership This follows from πM 2B F 2B T F which is the manufacturers profit under a single dealership given in 1427 144 Appendix The Legal Approach to Tying Section 3 of the Clayton Act passed in 1914 states that It shall be unlawful for any person engaged in commerceto lease or make a sale or contract for sale of goodsor fix a price chargedon the condition or understanding that the lessee or purchaser thereof shall not use or deal in the goodswhere the effect of such lease sale or contract for salemay be to substantially lessen competition or tend to create a monopoly in any line of commerce Since tying and bundling are frequently observed it is easy to infer that at least mixed tying is not illegal per se despite the fact that there have been several rulings on a per se basis against tying for interesting court cases see Asch 1983 and Gellhorn 1986 In fact the rulings against tying are associated with cases brought against firms that attempted to extend their monopoly power from one market to another which courts Page 389 term leverage as in subsection 1414 where we showed that tying may induce a consumer to purchase another product from his less preferred brandproducing firm and in rare case can cause a foreclosure of firms in the tied market Moreover the court ruled that the mere existence of a patent on a certain product does not entitle the patentee to impose a tiein on the purchaser of a patented product That is a patent holder of say a copy machine cannot impose on the buyer the use of its own brand paper In sum courts nowadays express the view that the plaintiff must show both that the producer maintains a monopolistic position in the tying product and that a tiein activity restrains a substantial volume of commerce of competitors in the tied product In such a case tying should be held as illegal per se Another issue related to tiein actually to mixed tying is its close relationship to price discrimination where consumers buying a tied package are priced differently than consumers who buy a single product In addition tying can serve as a tacit collusion between two firms producing complementary products Clearly if both firms fix their prices there is an immediate violation of antitrust laws but by using tying the firms are able to conceal the collusion Finally such tacit collusion may also serve as entrybarrier mechanisms 145 Appendix The Legal Approach to Vertical Restraints Section 1 of the Sherman Act passed in 1890 states that Every Contract combination in the form of trust or otherwise or conspiracy in restraint of trade or commerce amount the several states or with foreign nations is hereby declared to be illegal The complexity of the legality of vertical restraints lies in the fact that there is a wide spectrum of vertical arrangements and their relative success in enhancing manufacturers efficiency is open to debate A variety of cases and court rulings regarding several of these arrangements that are discussed in Asch 1983 and Gellhorn 1986 demonstrate the courts ambiguity about whether the rule of reason should be used in determining the legality of any given arrangement Until the late 1970s vertical price fixing and territorial restrictions were condemned under the per se rule as violations of the Sherman Act However several pricefixing arrangements were not always judged by antitrust laws since several states passed fair trade laws that also covered the issues of price maintenance Although vertical price fixing Page 390 is still per se illegal since the late 1970s and during the 1980s courts have expressed the view that the per se rule should generally not be used to evaluate vertical restraints in pricing because the rule violates the principle that manufacturers and dealers are free to establish the best arrangement for marketing their product In general it seems that courts have been more receptive to vertical arrangements that did not involve price restraints possibly because territorial restrictions would induce dealers to engage in providing more services and advertising Allowing several dealers to compete in a certain location would invoke the wellknown free rider problem where small dealers ride free on advertising by other dealers Thus some courts held that territorial dealerships are essential for promoting the product and for this reason a manufacturer has to insulate the dealer from competition since without promotion a manufacturer may lose the entire competition to manufacturers producing competing brands For this reason courts tend to use the rule of reason with respect to nonprice vertical restraints Finally courts had to deal with several cases of a refusal to deal where a manufacturer refused to deal with several retailers marketing the same product Although the right of a manufacturer to deal or not to deal is well established and reasonable the refusal to deal bears some similarity to exclusive territorial dealerships since refusal to deal is an effective punishment for those dealers who engage in pricecutting retailing That is revoking dealerships for price cutting can substitute a formal contract where a price fixing is explicitly mentioned 146 Exercises 1 Consider the mixedtying model studied in subsection 1413 and suppose that consumer 2 in Table 142 changes his or her tastes so that he or she attaches a value of 5 to each product That is Answer the following a Find the monopoly price of a package under pure tying b Find the monopoly product and package prices under mixed tying c Does the monopoly make a higher profit under mixed tying than under pure tying Prove it 2 Consider the market for textbooks analyzed in section 142 but suppose now that the publisher can make a commitment not to introduce a new edition in period 2 and that students believe the publisher when such a commitment is made Answer the following questions a What is the condition on F that will induce the publisher to make such a commitment Hint It is straightforward from equation 1417 Page 391 b Explain why the condition you found is less restrictive than the condition given in equation 1417 Hint This problem relates to the commitment problem of a durablegoods monopolist analyzed in section 55 3 Consider the market for textbooks analyzed in section 142 but suppose now that the maximum amount that the students are willing to pay for a new textbook is V the maximum they are willing to pay for a used book however is αV where 0 α 1 That is the secondperiod students are willing to pay less for a used textbook since a used textbook may contain marks and perhaps some missing pages We denote by the period t price of a new book and by the price of a used one Thus the utility of a generation t student is given by Answer the following questions a Suppose that the monopolist does not introduce a new edition in period 2 What is the maximum price at which the students would be able to sell used books Prove your result b Would this modification α 1 change the monopolys decision whether to introduce a new edition in period 2 4 Consider the singledealership problem analyzed in section 1431 Suppose that the manufacturer sells each unit to the dealer for d c unit manufacturing cost but in addition requires the dealer to pay a fraction of of the dealers profit Answer the following questions a Formulate the dealers profitmaximization problem under this contract and show that this contract maximizes the industry profit That is show that for any given φ the sum of the manufacturers and the dealers profit is equal to the profit made by a monopoly manufacturer selling directly to the consumer b How would your answer change if φ is the fraction of the enduser price instead of the dealers profit That is suppose now that the dealer pays a fraction of φ of the enduser price to the manufacturer for each unit it sells c Explain why shareinprofit types of contract are not frequently observed Hint Think of problems associated with having the manufacturer monitoring the dealers profit 5 Many home appliances stores in the United States advertise that those consumers who will trade in their old washing machine will receive a substantial discount on a new washing machine Explain why stores may find it profitable to engage in this sort of tradein Page 392 147 References Adams W and J Yellen 1976 Commodity Bundling and the Burden of Monopoly Quarterly Journal of Economics 90 475498 Asch P 1983 Industrial Organization and Antitrust Policy New York John Wiley Sons Benjamin D and R Kormendi 1974 The Interrelationship Between Markets for New and Used Durable Goods Journal of Law and Economics 17 381402 Burstein M 1960 The Economics of Tiein Sales Review of Economic Studies 42 6873 Carbajo J D de Meza and D Seidmann 1990 A Strategic Motivation for Commodity Bundling Journal of Industrial Economics 38 283298 Gellhorn E 1986 Antitrust Law and Economics in a Nutshell St Paul Minn West Publishing Co Horn H and O Shy 1996 Bundling and International Market Segmentation International Economic Review 37 5169 Lewbel A 1985 Bundling of Substitutes or Complements International Journal of Industrial Organization 3 101107 Liebowitz S 1982 Durability Market Structure and NewUsed Goods Models American Economic Review 72 816824 Mathewson G and R Winter 1986 The Economics of Vertical Restraints in Distribution In New Developments in the Analysis of Market Structure edited by J Stiglitz and G Mathewson Cambridge Mass MIT Press McAfee P J McMillan and M Whinston 1989 Multiproduct Monopoly Commodity Bundling and Correlation of Values Quarterly Journal of Economics 19 221234 Miller L 1974 On Killing Off the Market for Used Textbooks Journal of Political Economy 82 612620 Rust J 1986 When Is It Optimal to Kill Off the Market for Used Durable Goods Econometrica 54 6586 Seidmann D 1991 Bundling as a Facilitating Device A Reinterpretation of Leverage Theory Economica 58 491499 Tirole J 1988 The Theory of Industrial Organization Cambridge Mass MIT Press Whinston M 1990 Tying Foreclosure and Exclusion American Economic Review 80 837859 Page 393 PART V THE ROLE OF INFORMATION Page 395 Chapter 15 Monitoring Management Compensation and Regulation If you want something done right do it yourself Traditional adage Firms are organizations that are run and operated by people who use the technology to manufacture the products and then set quantity and prices to maximize profits in a given market structure The workers of the firm play the crucial role in controlling the production level quality and service to consumers Clearly since many firms are not owned by their employees a natural question to ask is what motivates workers and managers to devote efforts leading to increasing the firms profitability A second question immediately follows Suppose that the firms know what motivates the workers to work hard then given that the firm is a large and complex form of organization how can the firm reward its workers and managers if the relationship between an individual worker or manager and output cannot be observed The prevailing assumption is that workers are motivated by incentives that directly affect their standard of living However it should be pointed out that monetary incentives are not the only means by which to motivate workers In many cases workers derive satisfaction from having a sense of accomplishment from cooperation in achieving targets from making decisions and from developing production processes Page 396 and costreducing technologies Understanding the entity called a firm becomes even more complicated once we recognize that the different individuals in a given firm have different incentives therefore one way of modeling firms is to assume that a firm is a coalition of individuals with different interests This approach of course will not always coincide with tee assumption that firms are profit maximizers These types of motivations are hard to model and we therefore abstract other possibilities from this list and assume that workers and managers seek to maximize monetary rewards they receive from the firms they work for In this chapter we analyze problems facing managers and owners of firms who seek to maximize profits but are unable to fully monitor the efforts put out by their employees Rather than pursue the visualmonitoring solution we here attempt to develop economic mechanisms that would provide the workers with the monetary incentives to exert effort in their work We also discuss the firms cost of implementing these mechanisms We then analyze how governments regulate firms without knowing the precise production cost of the regulated firms What is common to all these problems is that the decision maker cannot observe what the workers managers or firms do Thus the decision maker is forced into devising economic incentives that would induce the workers managers or firms not to shirk work and reveal what is unknown to the decision maker In section 151 we analyze how a principal can provide an economic incentive to an agent that would induce the agent not to shirk work In section 152 we discuss a different incentive problemhow to induce workers sharing efforts in the same project to devote the optimal level of effort to it In section 153 we provide an explanation of why managers are really needed and shows that the separation of owners from managers can increase the strategic position of a firm In section 154 we provide one explanation of why firms pay according to rank rather than according to revealed output and why CEOs of large firms are paid astronomical salaries In section 155 we analyze a problem often faced by governmentswhether to approve and subsidize a project undertaken by an independent firm say a publicutility firm when the government does not know the exact production cost of the product or the service to be provided by the firm 151 The PrincipalAgent Problem The principalagent problem see Ross 1973 Grossman and Hart 1983 and Sappington 1991 for a survey exists in almost every social structure where some units are regarded as managers and some units as supervised agents This wellknown problem exists in every family where parents Page 397 not knowing whether their kids prepare their homework or not wish to reward the kids for good grades A landlord leasing his or her land to a tenant has no way of knowing whether yield is a product of the tenants working hard or shirking when weather has a large impact on the crops but he or she may wish to provide the tenant with a sufficient incentive to cultivate the land see Stiglitz 1974 A plaintiff expecting to win a large sum of money would like to encourage his hired attorney to work hard before a court appearance A school notobserving what professors actually do in class may wish to reward the professor according to the students achievements A government not observing the efforts of its workers would like to compensate its workers according to public polls on some aspects the government bureaucracy We first focus on firms consisting of two groups a manager and employees The manager hires a worker after examining the workers credentials which lead the manager to believe that the worker has the ability and the skill to perform the required task However even if the manager can ensure that the worker has the skills how can the manager be sure that the worker will indeed make the effort to use his or her skills That is what are the incentives for the worker to work hard and use his or her skills Well if the boss monitors the workers she can fire the worker when the worker does not work hard However if the boss monitors the workers she cannot do any other work which may jeopardize the entire operation of the firm Suppose that the manager decides to install TV cameras or hire supervisors to constantly monitor the workers movements It is clear that such an action would induce the workers union to take the manager to court for violating basic workers rights In addition monitoring is costly to managers and may or may not compensate for the extra output generated by monitoring For this reason in what follows we search for economic mechanisms that would substitute for the physical monitoring of workers Let us consider the following problem A restaurant owner the principal hires a waiter the agent to run the restaurant while she is away If the waiter does not work hard shirks customers will not get a proper service and consequently fewer customers will go to eat in this restaurant Therefore revenue will fan If the waiter works hard the restaurant becomes more popular and revenue will rise However given that the revenue is collected by the manager the worker may not have the incentive to work hard The timing of the interaction between the owner and the waiter is as follows First the owner designs the terms of the contract which specifies the payments the waiter will receive depending on the observed revenue of the restaurant The owner offers the contract to the waiter Page 398 and the waiter decides whether to accept the contract and start working or to choose some other work Second if the waiter accepts the contract then he goes to work and decides how much effort to exert in this work Finally the restaurants revenue is observed and the owner pays the waiter as promised in the contract Note that this commonly used setup implies that the owner is in control of the bargaining in the sense that she makes a takeitorleaveit offer to the waiter The waiter then can either accept the terms or reject them but is unable to bargain over the terms of the contract 1511 Providing economic incentives under Certainty An illustration The agent We denote by e the amount of effort put out by the agent We assume that there are only two degrees of effort that the agent can put out If the agent works hard he puts an effort level given by e 2 and if he shirks he puts an effort level given by e 0 We assume that if the agent does not take this job he can work at an alternative place say for the government and that the governments wage minus his effort yields a utility level of U 10 which is called the agents reservation utility Our agent likes money and dislikes work That is letting w denote the agents wage we assume that his utility function is given by The restaurant and the principal The revenue of the restaurant depends on the waiters effort level and is denoted by function Re If the waiter works hard e 2 the revenue is high and given by If the waiter shirks then the revenue is low so Altogether Finally the profit to the restaurant owner the principal is the restaurants revenue minus the wage bill which we denote by π That is Page 399 The contract The objective of the owner is to maximize the profit given in 153 We assume that the difference H L is sufficiently large so that the owner seeks to minimize the expected wage bill Ew while inducing the agent to work hard e 2 Which contract should she offer to the agent Obviously the contract has to depend on the revenue generated by the agents unobserved effort Let wH denote the wage rate that the principal promises to pay the agent when the revenue is H and let wL be wage paid to the agent when the revenue is L What should be the values of wH and wL that would maximize the principals profit subject to providing a sufficient incentive for the agent to work hard These incentives can be summarized by two constraints that the principal should consider while writing the contract Let us recall that the agent can work in another place and gain a net utility of 10 Thus in order to induce the agent to work hard in the restaurant e 2 the principal should write a contract specifying the values for wH in the event that R2 H and wL in the event that R2 L that would provide the agent with a utility level of at least 10 Thus in view of the agents utility function 151 the agents participation constraint is given by Even if the agent works for the restaurant the contract has to provide the agent with the monetary incentive to work hard That is the utility level generated by the net of effort income from working hard should be no less than the utility generated by shirking Thus the incentive constraint is given by Solving 154 for the equality case yields wH 12 Substituting into 155 yields wL 10 Finally the profit to the principal when e 2 is πH H wH H 12 and when e 0 is πL L wL L 10 Hence for this contract to be optimal for the principal we must assume that or What is wrong with this model The optimal contract that would induce the agent to work hard turned out to be very simple Since the utility loss to the agent from working hard is 2 the principal needs to pay the agent an additional 2 units of money to induce the agent to work hard Note that in this simple example lower values for wL would yield the same outcome since the agent has a sufficient incentive to work hard Page 400 The simplicity of this contract stems from the fact that the principal can actually monitor the agents effort by simply counting the revenue generated by the agents effort Thus this environment is the same as an environment with perfect monitoring In other words with no uncertainty the owner can calculate the precise effort exerted by the waiter by simply observing the restaurants revenue 1512 Providing economic incentives under uncertainty Let us consider the effects of the following events on the revenue collected by our restaurant A stormy day scares people away from going out an important football game is shown on TV or for some reason nobody is hungry on a particular day say because of National Diet Day These examples basically say that in addition to the waiters effort the revenue of the restaurant also depends on some other parameters We call these other causes the states of nature because they are beyond the control of either the agent or the principal Thus a high effort level put out by the agent cannot insure that the revenue will be high An increase in the agents effort level can only increase the probability for the event that Re H That is it is nature that determines the value of Re but our agent can affect the probability of each realization of Re by choosing his effort level Formally we assume that nature determines R2 and R0 according to Thus by working hard the agent can raise the probability of having R H from 04 to 08 Finally we need to slightly modify the agents utility function 151 to incorporate this uncertain environment Thus we assume that the agent maximizes his expected wage minus his effort given by where E is the expectation operator so in the present case Ew 08wH 02wL when e 2 and Ew 04wH 06wL when e 0 The participation constraint 154 is now given by reflecting the possibility that nature may play L despite the high effort put by the agent Page 401 The incentive constraint 155 is now given by That is the contract has to specify the agents statecontingent wages wH in the event that R2 H and wL in an event that R2 L that would yield a higher expected utility under e 2 than under e 0 Equation 158 implies that wL 604wH and 159 implies that wL wH 5 Altogether the optimal contract is wH 13 and wL 8 Monitoring versus economic incentives under symmetric information So far we have shown that economic incentives can substitute for the unpleasant physical monitoring of the agents actions That is we showed that the manager can achieve her productionservice goals without monitoring her workers at all provided that she knows how to write a contract that links the agents wage with the states of nature But our last question is at what cost does she achieve her goals Is the economic mechanism cheaper to implement than with the TV cameras or hired supervisors Note that hiring supervisors raises the problem of how to guarantee that the supervisors would have the incentives to catch the workers that shirk Well the supervisors would either have to be supervised or to be given the right economic incentives to catch those who shirk But who would then supervise the supervisors of the supervisors An important lesson to be learned from our example is that this economicincentive mechanism is efficient in the sense that it is not costly to implement To see that we compare the wage bill paid by the principal under the equivalent of perfect monitoring which is wH 12 and wL 10 with the nomonitoring under uncertainty wage bill wH 13 and wL 8 However notice that in the uncertainty case the expected wage bill is 08 13 02 8 12 which is identical to the wage bill the principal pays under monitoring Thus this economic mechanism is not too costly to implement and here hiring supervisors is not needed Finally the result that the expected wage bill under uncertainty is the same as the one under certainty is not a robust one because this result does not hold when the parties are risk averse That is the parties attach different probabilities to the realization of the states of nature the set H L in our case In the next subsection we will analyze how different attitudes toward risk affect the structure of the contract that the owner offers the waiter hence the owners expected wage bill and we will show that under this asymmetry the expected wage bill would exceed 12 Page 402 1513 Principalagent problem under asymmetric information The literature on the principalagent problem has been extended to analyze owners and waiters that have different attitudes toward risk In more professional terms the literature assumed that the owner and the waiters have different degrees of risk aversion We would now use a new concept called subjective probability that measures the probabilities each player assigns to the realization of the states of nature Thus we assume that both players acknowledge the same states of nature H and L but owing to their different backgrounds each player assigns different probabilities to the realizations Formally let us recall from 156 that the owner believes that RO2 and RO0 are realized according to Our modification of the previous model is that here we assume that the waiter believes that RW2 and RW0 are realized according to We would like to characterize the source of difference between the owner and the waiter We need the following definition Definition 151 Let there be two consumers denoted by i i 12 We say that consumer i is more risk averse than consumer j if when consumer j prefers a fixed sum of money over a lottery then consumer i also prefers the fixed amount In the present framework the waiter is more risk averse than the owner since the waiter is more skeptical than the owner about the realization of the good high state of nature That is the waiter attaches a lower probability to the H event and a higher probability to the L event The reason why we think of these differences in subjective probabilities as simulating different attitudes toward risk can be seen from the following illustration Suppose that the owner pays the waiter wH and wL where wH wL Then if the waiter exerts e 2 the expected wage bill for the owner would be which is higher than the expected wage received by the waiter Hence from the point of view of the waiter he values the expected wage bill Page 403 less than the owner reflecting the behavior that the waiter requires a greater compensation for working in an uncertain environment Finally the expectation operator E in the waiters utility function 157 should be interpreted as his subjective expectation which is different from that of the owners That is EWw 07wH 03wL if e 2 and EWw 04wH 06wL if e 0 The participation constraint 154 is now given by The incentive constraint 155 is now given by The participation constraint 1512 and the incentive constraint 1513 are drawn in Figure 151 The two constraints intersect at the point G Figure 151 Optimal contract under asymmetric information Any combinations of wH and wL to the left of the upward sloping curve 1513 are contracts that would induce the waiter to exert maximum effort Any combinations of wH and wL above the downward sloping curve 1512 are contracts that would be acceptable to the waiter Altogether it is clear that the owner would pick a contract which lies on the triangle above including the point G Finally the owner chooses a contract wH and wL to minimize the expected wage bill EOw she has to pay the waiter We remarked earlier that if H L is sufficiently large there is an equivalence between maximizing expected profit and minimizing expected wage Formally the Page 404 owner solves The isoexpected wage bill 08wH 02wL min EOw is also drawn in Figure 151 Note that this line is sloped 025 implying that the owners expected wage bill is minimized at point G Hence the owner would choose a contract given by wH 14 and wL 223 Clearly the waiter will accept this contract We conclude with the following proposition Proposition 151 The owners expected wage bill exceeds the waiters reservation utility plus his effort level Formally The intuition behind Proposition 151 is as follows Suppose that the waiter does not like to work under risky conditions Then in order to induce the waiter to work the owner must compensate the waiter for taking a random wage contract This compensation is reflected by the difference 1266 12 which is interpreted as the presto for being relatively more risk averse Finally note that the optimal contract bears some insurance for the waiter since from all individually rational contracts contracts located along the line 1512 in Figure 151 the ohooses the least risky contract In other words the contract that wH wL is minimized This insurance is needed since the waiter is more risk averse than the owner 152 Production with Teams The inability to monitor a workers effort also generates an inefficiency when the output of the firm depends on the effort levels of all workers assigned to work on a certain project which we call the jolt effort of a team This type of externality is commonly called the freerider effect in which a worker knowing that all other workers in a tern are putting a lot of effort into the project will have an incentive to shirk given that the group as a whole is rewarded on the value of the project that is when the individual workers are not rewarded according to their individual effort levels Consider a research lab developing the future product whose value is denoted by V In the lab there are N scientists workers who work on this project We denote by ei the effort put in by scientist i i 1 2 N Page 405 The value of the jointly developed product depends on the effort levels of all the N scientists and is given by That is equation 1515 can be viewed as a production function where the inputs are the efforts put out by the scientists Finally we denote by wi the compensation given to scientist i after the project is completed We assume that the value of the product is distributed to the workers so Σi wi V All scientists have identical preferences summarized by the utility function 1521 A digression Optimal effort levels Abstracting from the monitoring problem we suppose that each scientist can observe the efforts of his other colleagues and we suppose that they collude to maximize their utility levels We now wish to calculate what the optimal symmetric allocations of effort and output shares wages are and therefore we set ei e and wi w VN for every i 12 N If we substitute into 1516 the representative effort level e that maximizes a representative workers utility solves That is if the workers can theoretically collude observe each others effort and adjust their efforts to maximize their utility each should put out e 14 level of effort and the resulting total value would be 1522 The equaldivision economic mechanism Back to the reallife situation let us suppose that the manager of this firm rewards the scientists according to their equal share of the total value of output Formally let us suppose that the manager sets wi VN We look for a Nash equilibrium Definition 24 on page 18 in effort levels where each scientist takes the effort levels of his colleagues as given and chooses his effort level to maximize his utility 1516 Formally each worker chooses the effort level ei to Page 406 Therefore we can state the following Proposition 152 Under the equaldivision rule 1 If the team consists of a single worker the worker will provide the optimal level of effort That is if N 1 then en e 14 2 If the team consists of more than one worker each worker would devote less than the optimal level of effort That is if N 1 then en e 14 3 The larger the team is the lower will be the effort put out by each worker each would have a greater incentive to shirk That is as N increases en decreases Proposition 152 shows that offering the workers equal shares of the value of the output is insufficient to induce them to devote the optimal level of effort to their work So why not offer them a higher share of the output Well although it may be possible to induce them to work harder if all workers are offered a higher share of the output ie wi VN the total wage bill will exceed the value of output We now look at the effect of the size of the workforce on the total output as well as on the workers welfare level Substituting 1518 into 1515 for en yields that the Nash equilibrium value of output is Hence the difference between the optimal output level and the equilibrium output level is V Vn N 12 Substituting 1518 into 1516 yields that the Nash equilibrium utility level of each worker is For the sake of illustration we approximate N by a real number differentiating 1520 with respect to N thus yields that Hence Proposition 153 1 An increase in the number of workers on the team will increase the difference between the optimal output level and the Nash equilibrium output level That is V Vn increases with N Page 407 2 An increase in the number of workers will reduce the welfare levels of each worker That is UiN decreases when N increases Part 2 of the proposition shows that the freerider effect intensifies when the number of workers increases causing a further deviation from the optimal output level the optimal output level V N2 increases with the team size but the equilibrium level Vn 12 does not vary with the size of the team 1523 An economic mechanism that works Following Holmstrom 1982 we now discuss a rather tough incentive mechanism that would induce all the N workers to put forth the optimal effort level Suppose that the team sets the following rule If the team as a group achieves the optimal output level V then each team member receives VN If the teams output is different from V then all team members receive 0 Formally This mechanism makes each team member responsible for the entire output level of the team Under the equaldivision rule the marginal effect of each team member is lower than the marginal social value in this mechanism however the marginal value of each workers effort is the entire enterprise Is this the end Not exactly since this kind of allocation mechanism may suffer from a problem known in economics as time inconsistency That is this mechanism can work only if after each time the output is produced the manager fires and replaces all the workers However if workers continue to work on a new project it seems unlikely that workers would agree to let the manager confiscate all the output just because somebody has intentionally or unintentionally deviated from the optimal effort level Hence even if some deviation has occurred it looks as if the workers would be able to negotiate with manager or among themselves a redivision of the output given that some output has already been produced Since workers anticipate that the manager will renegotiate the contract the workers may not take this contract too seriously 153 Competition and Managerial Compensation Over this entire book we have always assumed that the players inside a firm share a common goal which is to maximize the firms profit Given this common goal it is clear that managers fulfill no economic goal Page 408 except perhaps to replace the owners who may be busy doing other business In this section we demonstrate that managers can play a role in a firm and that the separation between managers and owners who compensate pay the managers according to the goals they establish can increase the profit of the firm beyond the level achievable if the owners manage the firms by themselves Following Fershtman and Judd 1987 we analyze how managerial compensation schemes affect the firms actions which in turn affect the firms profit More precisely we examine the incentive contracts that principals owners of firms here will choose for their agents managers here What distinguishes this analysis from the principalagent analysis performed earlier is that here we analyze managerial compensation under a duopoly market structure As we demonstrate below it turns out that managerial compensation under duopoly is completely different from managerial compensation under monopoly for the very simple reason that under duopoly managerial compensation alters a firms strategic position Thus a firms owner can write a contract with a manager that may advance the firms strategic position beyond what could be achieved when the manager is instructed to simply maximize the firms profit In fact we show that profitmaximizing owners will almost never tell their managers to maximize profits It turns out that under Cournot duopoly competition each owner would want to motivate his manager toward a higher production level more sales so that the competing owners would instruct their managers to reduce their production level Let us consider a market for a single homogeneous product where the demand curve is given by p a Q p is the market price and Q is the aggregate quantity demanded There are two firms indexed by i i 1 2 Let qi denote the quantity produced by firm i and let Ri and πi denote the revenue and profit levels of each firm i The unit production cost of each firm is denoted by c where 0 c a 5c Thus recalling our Cournot market structure analysis of section 61 the revenue and the profit of each firm i i 1 2 are given by 1531 Incentives to managers The owner of each firm who could be a single person or the shareholders appoints a manager with an agreedupon compensation scheme Let Mi denote the compensation to manager of firm i and let us assume that the owner sets the compensation so that Page 409 Thus we assume that the manager of each firm i is promised payment of a fraction μi of a linear combination of the firms profit and the firms revenue For example if the owner of firm i sets αi 1 then the manager of firm i will simply maximize the firms profit and will earn a fraction μi of the firms profit In this case the owner will earn 1 μiπi In contrast if the owner sets αi 0 the manager will maximize the firms revenue and will earn a fraction μi of the firms revenue The purpose of this section is to demonstrate that owners will almost never set αi 1 meaning that owners will provide the incentive to managers not to maximize profits but instead to maximize a linear combination of profit and revenue 1532 A twostagedecisionlevel market game We assume that in the first stage the owner of each firm i chooses μi and αi to maximize the owners profit given by Thus the owner of each firm sets μi and αi to maximize the firms profit net of managerial compensation cost In the second stage the manager of each firm i takes μi αi and qj as given and chooses the output level of firm i qi Second stage Managers choose output levels For given qj and αi each manager chooses qi to maximize 1524 The firstorder condition and the bestresponse function of each firm i are given by Note that the bestresponse functions 1526 are slightly different from the response functions developed in the conventional Cournot model of section 61 since managers now are not maximizing profits alone The term αic2 means that the managers do not place a whole weight on the unit production cost since some weight is placed on revenue alone The bestresponse function of manager i is drawn in Figure 152 Figure 152 shows that if the owner of firm i lowers αi implying that the owner would like the manager to place a heavier weight on revenue than on profit the manager shifts his reaction function upward reflecting the fact that for every given qj firm i responds with a higher qi Solving 1526 yields the output level of each firm i and the aggregate industry output level as functions of the owners set control parameters Page 410 Figure 152 Bestresponse function of manager i when the owner lowers αi to α1 and α2 Hence We can easily see the effect each owner has on the industry output level Proposition 154 The industry aggregateoutput level increases when one or the two owners decrease their managers incentive to maximize profit Formally Q increases when αi decreases for some i i 1 2 Finally the equilibrium price as functions of αl and α2 is given by First stage Owners choose managers objective function Let us notice that the compensation parameter μi in the managers compensation scheme 1524 does not have any effect on the managers decision simply because managers incentives do not vary with scaling Mi up or down by a constant μi Hence owners can set μi as low as they wish assuming that managers have no alternative place to work Therefore to simplify our exposition we set μi 0 hence Mi 0 in the owners objective function 1525 A more general analysis would have to include a positive μi set according to the managers alternative salaries in competing industries The owner of each firm i takes μj and the output and price functions 1527 and 1528 as given and chooses αi that solves Page 411 The maximization problem 1529 is easy to solve if we observe that some terms in 1529 are not functions of α1 and α2 Therefore the solution to 1529 is identical to solving The firstorder condition yields the owners bestresponse functions The owners bestresponse functions are downward sloping implying that if the owner of firm 1 encourages his manager to place more weight on revenue than on profit reducing α1 the owner for firm 2 would respond by increasing the incentive for his manager to place a higher weight on profit than on revenue and hence to produce a lower output level Our main point can be demonstrated by the following experiment Suppose that firm 2 is an ordinary Cournottype firm managed by its owner and therefore only maximizes profit That is α2 1 in 1524 Substituting α2 1 into 1531 yields Proposition 155 Given that firm 2 only maximizes profit the owner of firm I will not maximize profit and will set α1 so that his manager will set the output level to equal the leaders output level see the Leader Follower model section 62 on page 104 Formally when α2 1 the owner of firm 1 sets α1 5c a4c hence by 1527 q1 a c2 The significance of Proposition 155 is that it demonstrates one possible reason why managers are needed That is by writing a contract of the type given in 1524 the owner of firm 1 can advance the strategic position of his firm beyond what could be achieved if the owner was managing the firm by himself Solving the two bestresponse functions 1531 yields the equilibrium incentive parameters given by Thus the equilibrium increases with the production cost parameter c meaning that when production cost increases the owners will induce their managers to place a heavier weight on maximizing profits When Page 412 the production cost is low owners will induce managers to place a higher weight on revenue maximization thereby increasing production levels Finally substituting 1532 into 1527 yields Thus Proposition 156 In an industry where the owners are separated from the managers firms output levels exceed the Cournot equilibrium output level derived in section 61 The significance of Proposition 156 is that the separation of managers from owners intensifies competition between the arms since owners design the managerial compensation schemes in a way that makes the managers more aggressive in sales Thus the separation of owners from managers reduces aggregate industry profit 1533 Collusion between the owners When the FTC investigates whether there is collusion between arms it is unlikely to look at managerial compensation as a source for collusion We now investigate whether owners can implicitly collude by setting the appropriate compensation schemes for their managers The interesting feature of this type of collusion is that managers will not even notice that a collusion to reduce output is taking place and thus will not have to be informed about it The framework developed above is very useful to investigate whether such an implicit collusion is profitable to owners Let us suppose that the owners collude by agreeing on how to compensate their managers and decide to set a common a into their managers compensation contracts Substituting α1 α2 α into 1530 yields that the owners choose α to maximize the joint profit given by The solution to the maximization of the joint profit is given by α a 3c4c Substituting into the managers output functions 1527 yields that For the purpose of this section we say that collusion occurs if the firms produce at levels below the Cournot output levels given by Proposition 157 Collusion among owners yields lower output levels and higher profit to each firm than under the Cournot competition Formally i 12 Page 413 154 Why Executives Are Paid More than Workers It is a common practice for firms to pay their executives much higher salaries than those paid to other workers Moreover these executivesworkers salary differentials do not seem to shrink even if the firm is not making a profit In other words large US firms do not seem to reduce the salaries of those in charge of the firm even if the firm does not perform very well Instead firms often fire the executive and replace him or her with another executive also paid a high salary An important lesson can be learned from this There is a common proposition that in a competitive market structure all employees workers and executives are paid the value of their marginal product This proposition implies for example that executives should be paid negative salaries when the firm loses some of its value a prediction that is never fulfilled Thus this proposition cannot explain salary differentials between executives and workers In this section we attempt to provide one explanation for why executives are paid much more than other workers Following Lazear and Rosen 1981 we show that firms may find it profitable to pay according to rank since large salaries of executives may provide incentives for all employees in the firms who with hard labor may win one of the coveted top positions In particular under imperfect monitoring where firms cannot observe employees effort levels eg section 151 we show that paying according to rank may provide workers with the incentives to exert effort at high levels For simplicity let us suppose that in a firm there are only two workers indexed by i 1 2 one of whom will be promoted and will become a managerexecutive We assume that promotion is granted to the worker who will turn over a higher output level We denote by qi the output level produced by employee i i 1 2 and assume that each employee can work hard by exerting an effort level ei e 0 or shirk by exerting an effort level ei 0 Then the relationship between the effort level of employee i and his or her output level is assumed to be given by Thus if the worker does not exert any effort ei 0 his or her output would be qi 0 However if the worker exerts high effort level ei e his or her output could still be qi 0 with a probability of 05 but could also be high qi H 0 with a probability of 05 Consequently when the firm finds out that a worker produced qi 0 it cannot infer whether Page 414 the worker was shirking or whether the low productivity is due to say bad weather or faulty equipment Let wE denote the wage rate the company pays to executives and wW the wage rate paid to other employees whom we call workers Thus if wE wW we say that executives are paid a higher wage than the workers Also suppose that one of the two workers will be promoted to the rank of an executive and that the worker who produces the higher output level will be the one that will be promoted In case both workers produce the same output levels we assume that promotion will be determined by tossing a fair coin thereby yielding a probability of 05 that each worker will be promoted Let p denote the probability that worker i 1 will be promoted Then Lemma 151 The probability that worker 1 will be promoted to a rank of an executive is given by Proof When el e2 e the event where q1 q2 H occurs with a probability of 14 Hence under this realization worker I is promoted with a probability of 18 Similarly worker 1 is promoted with a probability of 18 when the realization is q1 q2 0 Finally the event ql H 0 q2 occurs with a probability of 14 Summing up when e1 e2 e worker 1 is promoted with a probability of 12 The case where e1 e2 0 is identical to e1 e2 e When e1 e 0 e2 the event where ql H 0 q2 occurs with a probability of 12 Also the event q1 q2 0 occurs with a probability of 12 hence in this case worker 1 is promoted with a probability of 14 Summing up when e1 e 0 e2 worker 1 is promoted with a probability of 12 When el 0 e e2 the event ql q2 0 occurs with a probability of 12 hence in this case worker 1 is promoted with a probability of 14 Now let us assume that each worker i takes the effort level of the other worker as given and using the promotion probability described in Lemma 151 maximizes his or her expected utility given by We assume that first the firm sets its salary structure wW for the worker that is not promoted and wE for the promoted executive and then we Page 415 look for a Nash equilibrium Definition 24 on page 18 in the effort levels of the two workers Our major point is shown in the following proposition Proposition 158 1 If executives and workers are paid the same salary then no worker would put any effort into work Formally if wE wW then el e2 0 is a unique Nash equilibrium 2 If the firm pays the executive a sufficiently higher salary than what it pays the worker then both workers will put a high effort into their work Formally if the wage structure satisfies wE 4e wW then e1 e2 e is a unique Nash equilibrium Proof The first part it trivial since if winning a promotion is not followed by a salary increase no worker could gain by exerting effort To prove the second part note that wE 4e wW ensures that implying that worker 1 and similarly worker 2 will not deviate from a high effort level since the expected utility from working hard exceeds the expected utility from shirking Last we need to show that el e2 0 is not a Nash equilibrium This follows from implying that worker 1 and similarly worker 2 would deviate from shirking given that the other shirks Proposition 158 proposes an explanation of why executives are paid high salaries compared to other workers of the firm In principle it is clear that a firm should pay its chief executive officer an amount equal to his effect on the profitability of the whole enterprise Yet the costs of measurement for each conceivable executive are prohibitively expensive Instead it might be said that those in the running are tested by assessments of performance at lower positions Thus by running this rank tournament the firm would have a high probability of spotting the hard workers while inducing all workers to work hard Finally the model described in this section provides a reasonable explanation for the wage disparity between executives and workers in a given firm However the model cannot explain this wage disparity in those firms that tend to hire their executive from outside the firm Page 416 155 Regulating a Firm under Unknown Cost Often state government agencies are assigned to determine the price that public utility companies such as phone electricity and gas can charge their customers Under perfect information the regulating agency can simply set consumers unit price to equal marginal production cost and provide a lumpsum subsidy to cover for the fixed costs if any There is a large amount of literature on the regulation of firms and the interested reader is referred to Laffont and Tirole 1993 and Spulber 1990 In general the regulating agency does not know the production cost of the regulated firm Let us note that there are other important variables that are not known by the regulating agency such as the workers and managers efforts see Laffont and Tirole 1986 The regulating agency may require the regulated firm to report its production cost however under this situation of asymmetric information it is unlikely that the firm would report its true production cost That is knowing that the regulating agency will price the service by its marginal production cost the firm would have great incentives to overreport its production cost Following Baron and Myerson 1982 we propose an economic mechanism that would provide the firm with a sufficient incentive to report its true cost thereby enabling the regulator to mandate marginalcost pricing Consider an economy where consumer demand for phone services is given by p a Q where p denotes the price of a phone call and Q the quantity demanded There is only one phone company producing phone calls under a constantreturnstoscale technology Formally we assume that the firms unit output cost is given by c where c is known to the firm but not to the regulating agency We assume that the regulating agency conducts research on the cost of producing phone calls and finds out that the unit cost could be CH with a probability of ρ and cL with a probability of 1 ρ where cH cL 0 and 0 ρ 1 The firm itself is assumed to know whether it is a high or lowcost producer We assume that before the regulator acts the regulator receives a report from the firm indicating whether it is a high or a lowcost producer We denote the value of the reported cost by where We also denote by c the true value of the cost parameter which is known only to the firm Thus if the firm reports we say the firm is revealing the truth Otherwise we say the firm is lying about its cost structure The goal of the regulating agency is to maximize the expected value of social weftare which is defined as the expected sum of consumer surplus and the firms profit Page 417 Assumption 151 The instruments available to the regulating agency are 1 Mandating the market price as a function of the firms reported cost 2 Determining a lumpsum subsidy to the firm as a function of the firms reported cost 1551 Truthful revelation and the profit of the regulated firm We denote by the profit of the firm with a true unit production cost c that reports to have a unit cost of to the regulating agency Let us note that the firm may or may not report its true cost That is we can have it that or For every value of c and the firms profit is given by where and are the price and the subsidy mandated by the regulator as functions of the reported cost parameter We denote by πc the profit of the firm when it reveals its true cost parameter that is when Formally 1552 A mechanism that works We now characterize some useful properties that an economic mechanism should have Definition 152 An economic mechanism is said to satisfy the property called 1 incentive compatibility if the firm cannot increase its profit by not reporting its true cost parameter That is if for every cH cL 2 individual rationality if the firm makes a nonnegative profit when it is reporting its true cost parameter That is if In the present case a mechanism satisfies incentive compatibility if Page 418 and In other words if the firm happens to be a highcost producer then under this mechanism it cannot increase its profit by reporting to be a lowcost producer In addition if the firm happens to be a lowcost producer then under this mechanism it cannot increase its profit by reporting to be a high cost producer In the present case a mechanism satisfies individual rationality if That is by reporting the truth the firm will make a nonnegative profit We now state our main proposition Proposition 159 The following mechanism induces the firm to reveal its true cost parameter is incentive compatible individually rational and maximizes social welfare and all and satisfying Proof First note that this mechanism achieves the revelation of truth since scH and scL are nonnegative hence by 1541 the firm makes nonnegative profit under truthful revelation Also substituting pcH cH and pcL cL into 1539 and 1540 yields 1542 Hence this mechanism is incentive compatible and the firm will truthfully report its cost Now since the firm is reporting its true cost social welfare is maximized since consumers pay marginalcost prices Intuitively we can see that the regulator sets prices to equal the firms reported unit cost since marginalcost pricing is necessary for achieving the social optimum Then the regulator uses its lumpsum subsidy policy to induce the firm to report its true cost by making it profitable to report cH when the firm is a highcost producer and cL when it is a lowcost producer Moreover since firms in general would like to report that they are highcost producers in order to extract higher subsidies an optimizing regulator must offer the firm a higher subsidy when it reveals that it is a lowcost producer Formally equation 1542 shows the following Page 419 Corollary 151 In order to induce the firm to reveal its true cost the subsidy paid by the regulator to the firm must be higher when the firm reports a low cost than when the firm reports a high cost Formally scL scH Thus the regulator induces a lowcost producer to reveal its true cost by offering it a high subsidy This implies that in general regulators should reward efficient firms more than less efficient firms 156 Exercises 1 Consider the teammanagement problem studied in section 152 However suppose that the production function 1515 is now given by and suppose that the utility of scientist i 1516 is now given by That is our scientist now has an increasing marginal disutility from exerting efforts Answer the following a If scientists can collude and monitor each others effort calculate the optimal effort level that should be exerted by each scientist b Suppose that scientists cannot collude What is the Nash equilibrium effort level exerted by each scientist c Would an increase in the number of scientists intensify the freerider effect Prove your answer 2 Consider the managerialcompensation model analyzed in section 153 Suppose that firm 2 maximizes only profit but the manager of firm 1 is instructed to maximize a linear combination of profits and sales instead of revenue Formally assume that the manager of firm 1 is instructed to choose an output level q1 that solves Assuming that a c 1 answer the following a Solve for the output levels of firm 1 and firm 2 and demonstrate which firm produces a higher output level b Would the owner of firm 1 make a higher profit level if she follows firm 2 and maximizes only profit instead of a combination of profit and sales Prove your answer Page 420 157 References Baron D and R Myerson 1982 Regulating a Monopolist with Unknown Costs Econometrica 50 911930 Fershtman C and K Judd 1987 Equilibrium Incentives in Oligopoly American Economic Review 77 92940 Grossman S and O Hart 1983 An Analysis of the PrincipalAgent Problem Econometrica 51 745 Holmstrom B 1982 Moral Hazard in Teams Bell Journal of Economics 13 324340 Laffont J and J Tirole 1986 musing Cost Observation to Regulate Firms Journal of Political Economy 94 614641 Laffont J and 3 Tirole 1993 A Theory of Incentives in Procurement and Regulation Cambridge Mass MIT Press Lazear E and S Rosen 1981 RankOrder Tournaments as Optimum Labor Contracts Journal of Political Economy 89 841864 Ross S 1973 The Economic Theory of Agency The Principals Problem American Economic Review 63 134139 Sappington D 1991 Incentives in PrincipalAgent Relationships Journal of Economic Perspectives 5 4566 Spulber D 1990 Regulation and Markets Cambridge Mass MIT Press Stiglitz J 1974 Incentives and Risk Sharing in Sharecropping Review of Economic Studies 41 219255 Page 421 Chapter 16 Price Dispersion and Search Theory One should hardly have to tell academicians that information is a valuable resource knowledge is power G Stigler The Economics of Information The commonly agreed upon law of one price stating that identical products sold at the same location at a given time period must be sold for identical prices is actually rarely observed in any market Most retail markets are instead characterized by a rather large degree of price dispersion This chapter has two goals First to try to explain how such price dispersion can persist in markets where consumers behave in a rational manner that is when consumers search for the lowest price Second to explain how rational consumers optimally search for a low price in a market with dispersed prices Section 161 Price Dispersion demonstrates the possibility that information and search costs result in an equilibrium where a homogeneous product is sold at different prices Section 162 Search Theory analyzes how consumers optimally search for the lowest price in the presence of a price dispersion 161 Price Dispersion Prices of identical products often vary from one store to another So far we have managed to explain some differences in prices by product differentiation In section 73 we showed that a product sold in one location is actually a different product when it is sold in a different location in an economy where transportation is costly In section 141 we introduced another source of price dispersion which is a commonly used Page 422 marketing method of giving volume discounts or tying the sale of one product with the sale of another Such marketing methods would leave the impression that different stores charge different prices for identical items It is important to realize that all these observations do not imply that the law of one price is violated for the very simple reason that differentiated products are not homogeneous and therefore it is not surprising that they are not sold for the same prices Thus economists are still left with the challenge of how price dispersion can persist in markets where rational consumers search for lower prices In this section we attempt to explain price dispersion by introducing the cost of obtaining price information That is we assume that acquiring information on prices is costly to consumers and consumers always weigh the cost of searching against the expected price reduction associated with the search process For the literature on this topic see Pratt Wise and Zeckhauser 1979 Reinganum 1979 Salop 1977 Salop and Stiglitz 1977 Shilony 1977 Varian 1980 and Wilde and Schwartz 1979 There are many costs associated with searching for a lower price For example there is the cost of buying the appropriate newspapers and magazines More important the cost of the search is very high for individuals who have a high value of time those of us who earn a large sum of money for each additional hour of work Thus as we show below consumers with a high value of time will rationally refrain from searching for the information on lower prices and will buy the product from the first available store In contrast consumers with low search cost eg low value of time will find it beneficial to engage in a search in order to locate the store selling at the lowest price A model of search and price dispersion Let us consider an economy with a continuum of consumers indexed by s on the interval L H according to their cost for going shopping where we assume that H 3L 0 Thus consumers indexed by a high s s close to H are high timevalued consumers whose cost of searching for the lowest price is high The consumers indexed by a low s s close to L are low timevalued consumers for whom the cost of going shopping and searching for the lowest price is small Figure 161 illustrates how consumers are distributed according to their cost of shopping There are three stores selling a single product that is produced at zero cost One store denoted by D is called discount store selling the product for a unit price of pD The other two stores denoted by ND are expensive not discount stores and are managed by a single ownership that sets a uniform price pND for the two nondiscount stores Page 423 Figure 161 Consumers with variable search cost searching for the lowest price Define to be the average product price Formally We assume that the consumers do not know which store is a discount and which is expensive unless they conduct a search at a cost of s However consumers do know the average store price Thus if a consumer does not conduct a search he knows to expect that random shopping would result in paying an average price of Each consumer buys one unit and wishes to minimize the price he or she pays for the product plus the search cost Formally denoting by the loss function of consumer type s we assume that The parameter a measures the relative importance of the search cost in consumer preferences Clearly since each consumer s minimizes 162 a type s consumer will search for the lowest price if that is if the sum of the discount price plus the search cost does not exceed the average price which equals the expected price of purchasing from a randomly chosen store In contrast if then clearly buying at random is cheaper for consumer s than searching and buying from the discount store Definition 161 A price dispersion equilibrium is the prices and such that 1 The discount store cannot increase its profit by unilaterally deviating from the price 2 The owner of the two expensive stores cannot increase his profit by unilaterally deviating from the price 3 For every consumer s the consumer searches and buys from the discount store if and only if Otherwise the consumer buys from the first available store Page 424 It follows from Definition 161 that if some consumers search for the lowest price and some buy at random then there exists a consumer denoted by who is indifferent to the choice between searching and shopping at random Thus for the consumer indexed by we have Hence Consequently in view of Figure 161 for given prices pD pND all consumers indexed by pay the cost of s for searching for the lowest price and all consumers indexed by buy at random and pay an average price of The discount store We denote by EbD the expected number of customers shopping at the discount store To calculate EbD observe that pD pND implies that all consumers who search buy at the discount store simply because their search provides them with the knowledge of which store is discounting In addition on average half of the consumers who buy at random will randomly arrive at the discount store the lucky ones Hence since there are only two stores the expected number of consumers who shop at the discount store is given by The discount store takes pND as given and chooses pD that maximizes expected profit given by The firstorder condition is given by Hence the best response function of the discount store is given by The expensive store We denote by EbND the expected total number of customers shopping at the two expensive stores To calculate EbND observe that Page 425 consumers do not search and therefore buy at random Hence since there are only two stores the expected number of consumers who shop at the expensive stores is given by The owner of the expensive stores takes PD as given and chooses pND that maximizes expected profit given by The firstorder condition is given by Hence the bestresponse function of the owner of the expensive stores is given by Price dispersion equilibrium The bestresponse functions of the discount store 166 and the expensive stores 168 are drawn in Figure 162 The unique equilibrium Figure 162 The determination of the discount and expensive prices prices are found by solving 166 and 168 The consumer who is indifferent to the choice between searching and buying at random is then found by substituting the equilibrium prices into 164 Hence Page 426 Note that and that pD pND since we assumed that H 3L The following proposition is straightforwardly from 169 Proposition 161 An increase in the cost of search parameter α will increase the prices charged by all stores Also the difference in prices between an expensive store and the discount store pND pD increases with an increase in the search cost and declines to zero as the search cost becomes negligible The interesting conclusion that we can draw from Proposition 161 is that an increase search cost increases the monopoly power of both types of stores In contrast when search cost is negligible competition between the two stores intensifies and all prices drop to the competitive level zero in our case Thus search cost explains why different stores charge different prices by enabling the stores to differentiate themselves from rival stores by labeling themselves as discount or nondiscount thereby reducing competition According to 165 and 167 the expected number of buyers at each store is given by Thus the expected number of shoppers in the discount store is greater than the expected number of shoppers at an expensive store since the discount store attracts both informed and uninformed consumers whereas the nondiscount store attracts uninformed consumers only 162 Search Theory Our analysis so far has concentrated on how stores utilize consumer search cost in order to differentiate the consumers and charge them different prices In this section we do not analyze the stores but we assume that stores charge different prices Our goal here is to analyze how consumers behave in the presence of price dispersion More precisely we analyze how consumers with search costs conduct their shopping and how they determine how many stores to visit when searching for the lowest price This problem is faced by all of us When we go shopping we enter one store observe the price and ask ourselves should we proceed to visit another store Suppose we proceed with the search what guarantee do we have that the next store on our search list will have a lower price Page 427 Several authors have dealt with the consumersearch problem beginning with Stigler 1961 and more recently Lippman and McCall 1976 McCall 1970 and Rothschild 1974 For a nice exposition of these papers see Sargent 1987 Let us consider a city with n types of stores selling an identical product With no loss of generality we assume that the price charged by each store of type i i 1 2 n is pi i That is a store of type 1 charges p1 1 a store of type 2 charges p2 2 and a store of type n charges pn n Figure 163 illustrates the types of stores and the price charged by each type We assume that prices are exogenously given Figure 163 Prices in a consumersearch model and stores do not change prices That is the stores price determination process is not analyzed in this section so stores prices are taken to be given Let us consider a single consumer who visits stores for the purpose of finding the lowest price We make the following assumptions Assumption 161 1 The consumer knows the distribution of the n prices but does not know which price is charged by a particular store That is the consumer knows that in the market there are n prices ranged p 1 2 3 n but does not know the exact price charged by each individual store 2 The consumer searches sequentially The consumer bears a search cost of s 0 each time he or she visits a store Page 428 Assumption 161 describes a consumer who visits a store observes the stores price p and then has two options To buy the product for the offer p he or she has in hand or to continue visiting an additional store and pay an additional search cost of s This type of search is called a sequential search since the consumer can revise his or her action after each time he or she visits an additional store and receives a price offer for the product 1621 The reservationprice strategy Our consumer can potentially continue searching as long as he she likes Therefore each time the consumer visits a store he or she has to solve the same dynamic optimization problem since the price distribution is independent of time and the horizon is infinite Lippman and McCall 1976 showed that under this stationary framework the optimal strategy can be reduced to a myopic decision rule Let us suppose that our consumer visits a store and receives a price offer of p We define by vp the consumers expected price reduction from visiting one additional store while having a price offer p in hand Formally since each price is realized with probability ln In other words the gain from an additional search while having an offer p in hand is the expected price reduction from one additional search which is the expected gain from finding a price lower by one dollar p 1n plus the expected gain from finding a price lower by two dollars p2n and so on For example suppose that the consumer visits a store and receives an offer of p 3 Then what should be the expected gain from one additional search In this case The following lemma is a mathematical identity and is proved in the appendix section 163 Lemma 161 The sum of J numbers is given by Using Lemma 161 we have the following lemma Page 429 Lemma 162 The function vs defined by 1611 can be written as Proof By 1611 Then by Lemma 161 vp p 1p2n p2 p 2n Let us consider now the two options available to a consumer who is standing at a store after receiving a price offer of p If the consumer concludes the search by buying the product then his or her loss is p In contrast if the consumer rejects the price offer and searches one more time then the expected loss is the sum of an additional search cost s plus the current price offer minus his or her expected gain from searching one more time Formally the consumer with an offer p in hand minimizes Equation 1612 shows that a lossminimizing consumer would stop searching and buy the product whenever the price in hand satisfies Otherwise if p s p vp the consumer continues searching Hence Proposition 162 A consumer with a price offer p in hand will continues searching if the expected price reduction from one additional search exceeds the cost of an additional search Formally a consumer continues searching if and only if the price in hand p satisfies vp s A consumer behaving according to Proposition 162 is said to be using a reservationprice strategy Definition 162 A price is called a consumers reservation price if satisfies Figure 164 illustrates how the consumers reservationprice strategy is determined In Figure 164 a consumer enters a store and observes a price p If the consumer stops searching and buys the product on the spot However if the consumer observes a price then the consumer proceeds to the next store and buys or continues to another store depending on whether or In what follows we calculate the consumers reservation price From Definition 162 the reservation price is implicitly defined by Hence by Lemma 162 Page 430 Figure 164 Reservationprice strategy Therefore The solution to this quadratic equation yields Equation 1613 implies the following proposition Proposition 163 The consumers reservation price satisfies the following properties 1 If the search cost becomes negligible the consumer will continue searching until he or she is offered the lowest prevailing price Formally as 2 An increase in the consumer search cost s would increase the consumers reservation price 3 An increase in the number of stores charging higher prices ie increasing n would increase the reservation price Part 2 of the proposition states that when search costs increase the consumer is willing to purchase at higher prices in order to avoid additional search expenses Finally observe that we did not make an assumption about whether a consumer during the search can regret and return to an earlier store at no cost In the literature if a consumer can costlessly return to an earlier store for the purpose of buying at a price offered earlier he or she Page 431 is said to be performing a search with recall The following proposition explains why we did not bother discussing the issue of recall during our search analysis Proposition 164 Even if a consumer is allowed to costlessly return to stores that were visited earlier in the sequential search a consumer will never return to a store Proof Since an optimal search implies that the consumer employs a reservationprice strategy a consumer will always buy if he encounters a price satisfying and will never buy if Hence if a consumer did not buy at a store visited earlier in the search process it means that the store charged and a consumer has no reason to return to such a store 1622 The expected number of searches Given the price distribution and the consumers optimalsearch rule we now wish to calculate the expected number of stores the consumer will visit until the consumer finds a price lower than or equal to his or her reservation price We denote by σ the probability that a consumer will not buy when he or she randomly visits a store Since this search process is stationary does not vary with time σ is independent of time To find the value σ for a given reservation price let us note that the consumer never buys when he or her receives a price offer That is the consumer will not buy if Thus there are prices that exceed the cousumers reservation price Since each price has a probability of 1n to be realized the probability that a consumer will not buy at a store is We now ask what is the probability that a consumer buys the product in his or her first store visit Clearly the probability of buying is 1 σ What is the probability that the consumer buys the product in his or her second store visit Clearly the probability that the consumer does not buy in the first store is σ and the probability that he or she buys in the second visited store is 1 σ Hence the probability that the consumer does not buy in the first store and buys in the second store is σ1 σ because the price distribution is time independent What is the probability that the consumer buys the product in his or her third store visit The probability that the consumer does not buy the product in the first and second visited stores is σ2 Hence the probability that he or she buys in the third store is σ21 σ Page 432 Finally what is the probability that the consumer buys the product in his or her ts store visit Clearly the answer is σt11 σ To find the expected number of stores to be visited before buying the product we need to sum the probabilities of buying at each given visit times the visits number Formally the expected number of store visits denoted by μ is given by Equation 1615 can be simplified using Lemma 91 which is proved in section 99 Hence Equation 1616 states that the expected number of stores to be visited by our consumer equals one over the probability that he or she buys in a single store visit 163 Mathematical Appendix Proof of Lemma 161 Let φ denote the sum and consider the following sum Since each column sums up to J 1 and there are J columns we have it that 2φ JJ 1 Hence φ JJ 12 164 Exercises 1 Consider the pricedispersion model developed in Section 161 a Show that if the search cost becomes negligible for some consumers then there will not be a discount store Hint Analyze what happens to the equilibrium market shares and prices when b Show the same for the case where search costs are uniformly low Hint Consider the case where α 1 2 Consider the consumers optimalsearch model analyzed in section 162 Suppose that there are nine types of stores each selling at a different price drawn from a uniform distribution where Answer the following questions a Construct a table showing the consumers reservation price and the expected number of store visits under different values of the Page 433 search cost parameter More precisely consider the cases in which s 0 1 2 3 4 5 b What is the value of s that will cause the consumer to purchase the product at his or her first store visit c What is the value of s that will cause the consumer never to buy the product unless the price is p 1 d Using the value of s that you found in subquestion c calculate the probability that the consumer will search forever Prove and explain your result 165 References Lippman S and J McCall 1976 The Economics of Job Search A Survey Economic Inquiry 14 347368 McCall J 1970 Economics of Information and Job Search Quarterly Journal of Economics 84 113126 Pratt J D Wise and R Zeckhauser 1979 Price Variation in Almost Competitive Markets Quarterly Journal of Economics 93 189211 Reinganum J 1979 A Simple Model of Equilibrium Price Dispersion Journal of Political Economy 87 851858 Rothschild M 1974 Searching for the Lowest Price When the Distribution of Prices is Unknown Journal of Political Economy 82 689711 Salop S 1977 The Noisy Monopolist Imperfect Information Price Dispersion and Price Discrimination Review of Economic Studies 44 393406 Salop S and J Stiglitz 1977 Bargains and Ripoffs Review of Economic Studies 44 493510 Sargent T 1987 Dynamic Macroeconomic Theory Cambridge Mass Harvard University Press Shilony Y 1977 Mixed Pricing in Oligopoly Journal of Economic Theory 14 373388 Stigler G 1961 The Economics of Information Journal of Political Economy 69 213225 Varian H 1980 A Model of Sales American Economic Review 70 651659 Wilde L and A Schwartz 1979 Equilibrium Comparison Shopping Review of Economic Studies 46 543553 Page 435 PART VI SELECTED INDUSTRIES Page 437 Chapter 17 Miscellaneous Industries Every industry is special The author In this last chapter I would like to emphasize the point that there is no single model that can be applied to the analysis of all industries Each industry has different characteristics such as different consumers tastes for the product or service and different technologies for producing the relevant products or services Thus despite the fact that there are general modeling techniques such as the commonly used market structures developed in the first and second parts of this book it is my view that each market phenomenon has to be explained in a specific ad hoc model In other words the procedure of borrowing models from one market to explain a different market generally does not work well To emphasize the need for unique modeling techniques we analyze three types of markets here that we regard as special This does not mean that the industries analyzed so far in this book are less than special rather the markets analyzed here simply did not fit any category developed earlier Section 171 Restaurant Economics analyzes a wellknown observation in which prices often do not rise in the presence of excess demand Section 172 Airline Economics analyzes an industry in which in addition to prices and quality the airlines route structure can be used as a mechanism to raise profits Section 173 Tragedy of the Commons describes a wellknown problem in which firms use scarce factors of production that are public properties Section 174 Congestion provides an economic theory to resolve traffic congestion problems Page 438 171 Restaurant Economics We can observe with some astonishment that popular restaurants theaters bars and dancing places often have people standing in line to get in What is even more astonishing is that these entertainment places do not raise prices in the presence of queues excess demand as predicted by the simple conventional supplyanddemand theory That is simple supplyanddemand theory tells us that in the presence of excess demand a firm can increase its price without reducing its output level thereby increasing its profit So why do restaurant owners refrain from raising prices when they observe the formation of lines front of their establishments It turns out that restaurant economics has a lot in common with the economics of compatibility and standardization described in chapter 10 Restaurants relate to the theory of compatibility in that the demand for restaurants by a certain consumer is affected by social conditions that are in turn affected by the restaurant choice of other consumers Hence the demand for some restaurants coffeehouses nightclubs discotheques and other entertainment and sports clubs exhibit network externalities 1711 A restaurant monopoly model Becker 1974 1991 proposes a solution for this puzzle Becker argues that the demand for entertainment places differs from the demand for oranges because social interactions affect the demand for restaurants but not the demand for oranges In the language of chapter 10 the preferences for entertainment places exhibit network externalities thereby generating demand curves that are not always downward sloping Figure 171 illustrates a possible demand facing a popular restaurant where Q denotes the number of customers and p the price of a meal In Figure 171 the demand is downward sloping at low demand levels reflecting a behavior that when there are few visitors in the restaurant the social effects are insignificant so quantity demanded responds to price in the usual fashion At certain demand levels the demand is upward sloping reflecting the behavior of being in so that the customers are willing to pay more as the number of customers increases At the demand level associated with the price pmax the restaurant gets so crowded so that consumers will increase the demand only if price falls The supply side is fixed by the number of tables in the restaurant the restaurant cannot supply more than QH meals at a given time or in the case of theaters there is always a limited seating capacity The corresponding market clearing price is denoted by pe At this price there are two equilibria one in which the quantity demanded QH equals the Page 439 Figure 171 The equilibrium restaurant price restaurants capacity and the other in which the quantity demanded is low We first would like to know how a monopoly restaurant prices its meals when facing the demand curve given in Figure 171 Proposition 171 A unique restaurant monopoly profitmaximizing price is given by pm pmax At this price the monopoly restaurant will face an excess demand queues for its meals measured by ED in Figure 171 Note that Proposition 171 may not hold if the demand function is very inelastic in the neighborhood of QL Proof If the monopoly sets pm pmax then the number of customers does not increase since the restaurant cannot sell beyond its capacity level QH Hence it is not profitable for the monopoly restaurant to reduce the price of its meal If the monopoly slightly raises its price the unique equilibrium quantity demanded drops to QL Hence the sharp discontinuous decrease in quantity demanded would make a price increase not profitable for the monopoly restaurant 1712 Extensions and discussion of the restaurant model The model discussed in the previous subsection raises two questions about the generality of the model in terms of the assumption placed on the demand structure and on the market structure Page 440 The demand side The demand function displayed in Figure 171 is an aggregate demand function portraying the behavior of a group of consumers A natural question to ask is what kind of heterogeneous consumer preferences would generate a demand curve similar to the one in Figure 171 Karni and Levin 1994 provide an example of a group of consumers who have different preferences toward their ideal restaurant size and develop the aggregate demand curve given in Figure 171 Competition between restaurants Although the model of the previous subsection predicted that a monopoly may refrain from raising the price even in the presence of excess demand the model does not explain Beckers main observation that two restaurants serving identical food at similar prices may be faced with a situation wherein one restaurant has empty seats while the other has long lines of hungry customers Formally the question is how would an equilibrium look if two restaurants compete in prices in this market Karni and Levin show that a Nash equilibrium for this game does not exist but a Leader Follower equilibrium may exist However as they point out there is no good reason why one restaurant would behave as a leader and the other as a follower Finally Conner and Rumelt 1991 provide another application for this socially induced upward sloping demand curve They develop a model of software piracy that shows a demand curve similar to that portrayed in Figure 171 the demand increases with the number of users buyers and thieves using the same package of software They show that a software firm may increase its profit by lowering the protective measures installed into the software say by removing protective plugs since an increase in the number of users that steal this software may boost the demand by honest buyers 172 The Airline Industry You may go to heaven or hell when you die but youll certainly stop in Atlanta hub airport on the way Folk saying in Florida You may go to heaven when you die but at least its a hell of a lot cheaper than going to Atlanta Denied by IATA Transportation services are different from other services or products in that they are not provided at a fixed location a transportation service Page 441 begins at a certain city of origin and ends at a different location called the destination However even if the points of origin and destination are well defined transportation services can be differentiated by different routings that connect origins with destinations That is airline firms or bus companies can transport passengers via different cities or just provide direct nonstop services yielding different costs of operation to firms and different levels of satisfaction to consumers The object of this section is to analyze the effects of route or network structuring on the profit of airline firms as well as on consumer welfare Our major observation of network restructuring comes from the recent deregulation of the US airline industry see Borenstein 1989 and Viscusi Vernon and Harrington 1992 for the effects of the US deregulation Perhaps the most visible outcome of this deregulation is the increased use of the hubandspoke HS network That is the increase in the competition among airline firms has caused airline firms to decrease the relative number of nonstop direct flights and to reroute passengers via a third city which we call a hub The HS is also very common in the overnightpackagedelivery industry in which small packages are flown to a single City hub and from there planes leave for all destination points In this section we demonstrate that a unique feature of transportation firms is that in addition to setting prices or quantities airline firms use network structuring as a strategic variable For the sake of illustration we break the analysis into two extreme demonstrations In subsection 1721 we analyze the effect on the airline firms cost of operation of altering the network from direct flights to HS In subsection 1722 we analyze the effect of this alternation on consumer welfare and airline pricing Figure 172 illustrates a tricity environment where there are three cities denoted by A B and C Figures 172ac illustrate fully connected networks FC where all passengers fly nonstop from origin cities to their destinations Figures 172bd illustrate hubandspoke networks HS where all passengers except those whose city of origin or destination is city B fly indirectly and stop at the hub city B In Figures 172ab we illustrate a oneway environment where there are n1 city A passengers wishing to travel from A to B and additional n3 passengers wishing to travel from A to C In addition n2 city B passengers wish to travel to city C whereas city C residents do not like traveling and therefore wish to stay at home It turns out that the oneway environment is analytically similar to the more general roundtrip environment illustrated in Figures 172cd We therefore abstract from the roundtrip environment and focus our analysis on the oneway environment illustrated in Figures 172ab Also Page 442 Figure 172 Fully connected FC and hubandspoke HS networks we do not analyze competition among airline firms and therefore focus on a single monopoly airline firm providing services to all passengers in all cites 1721 The cost approach Several economists claim that due to the topographical network structure imbedded in transportation services airline firms have technologies in which the cost functions are affected not only by the number of passengers but also by the network structure see Bittlingmayer 1990 Let the total cost of an airline be a function of the number of passengers transported on each route and denote it by TCn1 n2 n3 Definition 171 An airline technology is said to exhibit economies of scope if that is if the cost of operation of a firm operating on all the three routes is lower than the sum of costs of three individual firms each operating on a single route Page 443 Definition 171 is given here only for the sake of illustration and is incomplete since the property called economies of scope generally implies that the function TC satisfies a property more general than the one given in Definition 171 called subadditivity For a rigorous treatment of economies of scope see Baumol Panzar and Willig 1982 Panzar 1989 and Sharkey 1982 Now let us suppose that there is only one airline serving the three cities Which network of operation will be chosen by the airline firm Would it operate an FC network or an HS network Let TC be a separable cost function defined by where Under the FC network the total cost of operation is TCFC 3F n12 n22 n32 where under the HS network TCHS 2F n1 n32 n2 n32 Assuming equal number of passengers on each route we have it that That is if the fixed cost associated with maintaining a route route 3 is large relative to the number of passengers on each route then the HS network is the costsaving network If the fixed cost of operating a route is small F is small then the FC becomes the costsaving network of operation Alternatively the HS is less costly to operate when there are fewer passengers This is part of the reason why the recent increase in competition due to deregulation caused most airlines to shift to HS networks 1722 The passengers demand approach In this subsection we analyze the polar case of subsection 1721 and consider the demand effect of establishing the network structure Following Berechman and Shy forthcoming we redefine the output of an airline firm to be the frequency of flights number of departures per day or week instead of the number of passengers flown on each route Using frequency as the measure of an airlines output instead of the conventional measure of passengermile has two advantages First the cost of an airline depends on the number of departures per unit of time and less on the number of passengers boarding each departing aircraft second passengers utility is greatly influenced by the number of departures per Page 444 unit of time since a higher frequency of service implies a shorter waiting time for passengers The airfare per trip on mute i is denoted by pi Let di 01 denote whether a flight is a direct one di 1 or not di 0 On each route i each of the ni passengers is assumed to travel only once The utility of a passenger on route i is affected by the fare pi by the frequency of flights fi and by whether the flight is direct or indirect Formally the utility function of a passenger on route i Ui is given by where δ is the extra dollars a consumer is willing to pay for a direct flight represents consumers utility gain from frequency Thus consumers preferences exhibit diminishing marginal utility of frequency on route i Although in this model each consumer travels only once during a given time period frequency is still important to consumers simply because it allows them greater flexibility in commuting and in saving time Finally let us assume that the airline firms cost of operation is c 0 per departure Direct flights The fully connected network FC Let us suppose that all flights are direct Then the monopoly airline extracts all consumer surplus by setting By symmetry all routes will be served with equal frequency hence we set f fi for every i Thus the single airline chooses fd that solves where πd denotes the profit of a monopoly firm providing service on an FC network The firstorder condition is given by Clearly Hence the monopolys frequency price and profit levels on each mute i i 123 are given by Hence Proposition 172 When there is a single firm operating an FC network then 1 the profit maximizing frequency increases exponentially with the number of passengers on each route n and decreases exponentially with the cost per departure c Page 445 2 the airfare and the profit level increase with the number of passengers n and passengers willingness to pay for a direct service parameter δ Since passengers are assumed to be willing to pay more for a more frequent service an increase in the number of passengers on each route would increase the airlines revenue generated by providing a higher frequency of service This explains why airline firms may choose to operate aircraft with less than full capacity together with a higher frequency of service rather than with fully loaded aircraft with a lower frequency of service The hubandspoke network HS Suppose now that the monopoly airline does not provide a direct flight on route i 3 but instead transports all passengers via a hub at city B Since now passengers on route 3 travel indirectly d3 0 we have it that whereas i 12 Still assuming that ni n for all i the airline chooses a common frequency f to maximize total profit Thus where the superscript h denotes a variable under the HS network The firstorder condition is given by Also Hence under the HS network the monopolys frequency on each served route prices and profit are given by Since the US deregulation it has been observed that airfares to hub cities are relatively high From equation 176 we can state Proposition 173 Under the HS network the monopolys airfare for flights originating or ending at the hub city exceeds the airfare paid by passengers whose destination is not the hub city Formally This result emerges because passengers value a direct flight more than an indirect one Consequently there is a lower surplus that the monopoly airline can extract from those passengers who fly indirectly thus the airline must charge them a lower fare Page 446 A single airline A comparison of FC with HS networks We now compare the monopolys frequency number of passengers per flight and prices on each route under the FC and the HS networks A comparison of equations 174 with 176 yields the following proposition Proposition 174 1 A monopoly airline will operate with greater frequency under the HS network than under the FC network Formally fh fd 2 The airfare set by the monopoly for passengers who start or end their trip at the hub city B is higher under the HS than under the FC Formally for routes i 1 2 3 If passengers valuation of direct flights δ is higher than a critical value the airfare for passengers traveling from city A to a nonhub destination at city C is lower under the HS than under the FC network Formally there exists δ δ n4c such that for every δ δ We now ask under what conditions a monopoly airline firm would switch from an FC to an HS Comparing the profit levels 174 with 176 yields the following proposition Proposition 175 The monopoly airline will operate an HS network as long as passengers valuation of direct flights is less than a threshold value that is δ δ where δ 3n8c Otherwise it would operate an FC network 1723 Should airfare be regulated From time to time it is argued that the regulator the Civil Aeronautics Board or CAB in the case of aviation should set a minimum airfare in order to prevent stiff competition among airline firms Let us also note that international cartels such as IATA also attempt to set minimum airfare for some routes Posner 1975 suggests an easy method for evaluating airfare regulation Figure 173 illustrates the market for air transport on a certain route Suppose that all the airline firms have identical per passenger cost of c0 Clearly under airfare competition the equilibrium airfare would drop to p0 c0 Now suppose that the CAB sets a minimum airfare of pmin c0 for the purpose of helping the airline firms earn abovenormal profit However under such price floor Posner observes that the following will happen Page 447 Figure 173 Evaluation of airfare regulation Proposition 176 Given that airline firms are not allowed to compete in prices and that they all charge airfare equals to pmin the airline firms will compete in service food drinks frequency etc Hence nonprice competition will increase the airline costs until effective per passenger cost is raised to c1 pmin Consequently the regulators minimum airfare regulation will not raise the profit of the airline firms above the normal level The important lesson that follows from Posners observation is that the regulator cannot stop competition among airline firms since airfare competition would be replaced by service competition What remains to check is the welfare effect of minimum airfare regulation Figure 173 illustrates that with the higher airfare passengers would reduce their flights to Q1 Q0 Hence consumers surplus see subsection 323 for a definition would drop and since the airline firms make zero profit before and after the regulation social welfare must drop as a result of regulating the airfare Well our argument is still incomplete since we neglected to take into account the possibility that consumers welfare may rise because the airline provides a better service In fact a better service may induce more people to travel by air so the demand would shift to D2 in Figure 173 Now we are left with the question of by how much welfare and demand could increase as a result of the better service Posner argues that the increase in the demand should be very small since otherwise the airline firms would compete in service even when price competition is present This is a very important logical argument stating that the absence of competition in service before regulation implies that the increase in service associated with a regulated higher airfare must be welfare reducing Putting it differently we can say that the passengers are forced to pay Page 448 a price for the service that is higher than their valuation of the service Note that the data confirm this argument since in practice the main concern of most passengers is the price or mileage accumulation which can be translated to price cuts and not the service For this reason inflight service after the US aviation industry was deregulated in 1978 was reduced to a minimal level 173 The Fishing Industry In this section we analyze industries that use public resources as factors of production The most notable example are the resources found in international waters beyond narrow bands of the sea belonging to specific countries Such resources include a variety of seafood and offshore oil Another applicable example is livestock grazing over public land The problem arising from the use of common properties as factors of production stems from the fact that commonproperty factors of production are not sold in competitive markets Hence the economic factor prices that should reflect their relative scarcity do not play a role in firms profit maximization problems since firms behave as if the cost of obtaining public factors is zero Hence the tragedy of the commons arises from overuse of these factors This wellknown problem is analyzed in several papers for example Coase 1960 Comes and Sandler 1983 Haverman 1973 and Weitzman 1974 Let us consider an economy with n fishermen Let hi denote the hours of fishing devoted by fisherman i i 1 n and H the aggregate fishing time devoted by all the n fishermen Formally We denote by the aggregate number of hours devoted to fishing by all fishermen except fishermen i Formally The aggregate weight of fish collected by all fisherman together is denoted by Y We assume that the catchoffish production function is given by This fishing production function exhibits decreasing returns in the sense that each additional hour allocated for fishing results in a smaller catch Page 449 than the previous allocated hour In addition this production function exhibits an externality since the marginal product of each fisherman depends on the amount of hours devoted by all fishermen Thus the more hours put in by any fisherman the lower the productivity of each fisherman We denote by yi the catch of fisherman i and assume that the share of fisherman i in the aggregate catch depends on the share of time devoted by fisherman i relative to the aggregate fishing time Formally the catch of fisherman i is given by Thus the catch of an individual fisherman i is an increasing function of his or her own effort and a decreasing function of the aggregate effort reflecting the fact that due to the decreasing returns the marginal product of fisherman i decreases with the aggregate fishing time We normalize the price of one ton of fish to equal 1 and denote the wage rate per hour of fishing by w w 0 We assume that there are no other costs associated with fishing and that the use of public national and international water is free of charge 1731 Oligopoly equilibrium in the fishing industry The only strategic variable available to each fisherman i is the amount of time to be allocated for fishing hi We look for a Nash equilibrium see Definition 24 on page 18 in fishing time allocation among all the n fishermen Formally each fisherman i takes the amount of time allocated by other fishermen as given and chooses hi that solves The firstorder condition for 1710 is given by It can be easily verified that We look for a symmetric Nash equilibrium where each fisherman invests the same amount of effort into fishing We denote the common equilibrium effort level by he for all i 1n Hence and He nhe Substituting these values into 1711 yields Page 450 Substituting 1712 into 177 and 179 yields the aggregate and individual catch From 1712 we have the following proposition Proposition 177 1 An increase in the number of fishermen would increase the aggregate fishing time but would decrease the fishing time of each individual fisherman Formally as n increases He increases but he decreases 2 An increase in the number of fishermen would increase the aggregate fish catch but would decrease the catch of each individual fisherman Formally as n increases Ye increases but he decreases 3 An increase in the price of the nonpublic factor would decrease both the effort and catch of each fisherman and decrease the aggregate industry catch Formally an increase in w would decreases he Ye and 1732 The social planners optimal fishing We now investigate from a social viewpoint whether the n fishermen are engaged in too little or too much fishing Let us suppose that the social planner is endowed with the power of granting hours of fishing to the n fishermen in the industry We denote by h the common allocation of fishing hours to each fisherman Letting denote thee aggregate amount of fishing hours allocated to the entire industry the social planner chooses H that solves The firstorder condition yields Hence Therefore Proposition 178 1 The aggregate amount of fishing hours devoted by an oligopolistic fishing industry exceeds the socially optimal level Formally for any He H Hence the fishing industry is overproducing That is Ye Y Page 451 2 The deviation between the socialist optimal amount of fishing and the oligopolistic fishing level increases with the number of fishermen in the industry Formally HeH hence YeY increases with n Let us note that when there is only one fisherman n 1 He H since the monopoly internalizes the externality and fishes at the optimal level Levhari and Mirman 1980 analyze a simple dynamic fishingcompetition model with variable fish population and demonstrate that the overfishing problem extends also to dynamic competition between countries 1733 Fishing licenses and taxation The social planner has two policy tools that can partially or completely correct the deviation between the equilibrium industry amount of fishing and the socially optimal fishing level Let us suppose that the social planner cannot tax the fishermen but has the authority of granting fishing licenses thereby controlling the number of fishermen in the industry We investigate the effect of granting licenses on aggregate output in the following corollary to Proposition 178 Corollary 171 Limiting the number of licenses will reduce the deviation of the industry output from the socially optimal output level More precisely granting a single license will make the equilibrium fishing level equal to the socially optimal level Formally when n 1 He H Granting a single license cannot be an optimal solution if consumer welfare not considered in the present analysis is taken into account However in what follows we show that taxation can bring the industry to meet the socially optimal level of production Let us suppose that the social planner taxes the fishermen on each hour of fishing Formally let t denote the fee each fisherman has to pay for each hour of fishing Clearly the effect of this tax is similar to that of increasing the wage per hour of fishing that is the total cost of hi hours of fishing is now given by TCihi w thi i 1n Hence from 1712 we can conclude that the industrys total fishing time when a tax of t is imposed on each hour of fishing is given by Denoting by t the tax rate that would induce the industry to fish at the socially optimal level t is determined by solving Page 452 Hence Consequently Proposition 179 1 The optimal tax per hour of fishing increases with the cost of fishing and the number of fishermen in the industry Formally t increases when w and n increase 2 When there is only one fisherman then the optimal fishing tax is zero Formally when n 1 t 0 174 Public Roads and Congestion So far in this book we have not analyzed a commonly observed externality type called congestion We define congestion as a social interaction where the participation of each individual slows down the service received by other consumers The reader probably does not need to be convinced about this observation since congestion is found in every aspect of our life Highways in major cites are congested during the day in the sense that traffic moves at a slow pace Telephone lines are busy during peak time Air traffic controllers impose delays on departing aircraft when they feel that they cannot comply with requirements for aircraft separation as demanded by the Federal Aviation Regulations Let us consider N passengers who work in a downtown of a major city and wish to be transported from the suburbs to downtown every morning There are two possible methods for getting downtown Each passenger can ride a train or can drive a car Let tT denote the travel time of the train and tC the travel time in a car We normalize the travel time of the train to tT 1 one hour The driving time to downtown depends on the traffic congestion and therefore depends on the number of all passengers who decide to drive a car Formally let the driving time be given by The parameter α measures the driving time that is independent of congestion such as the time it takes to start and heat a car to check the oil and so on The parameter β measures the effect of congestion on travel time which depends on the quality of the highway the number of lanes and traffic lights Page 453 We denote by v the value of time by nT the number of passengers who ride the train and by nC the number of passengers who drive their cars where nC nT N Suppose that the train operator is competitive so the train ticket equals the unit cost which is denoted by φ Altogether the monetary value of the loss to a passenger who rides the train is given by The monetary value of the loss to a passenger who drives a car is given by 1741 Equilibrium highway congestion We assume that there is a large number of passengers wishing to go downtown so each passenger ignores his or her marginal effect on congestion Hence each passenger takes nC as given and minimizes Therefore if in equilibrium passengers use both transportation methods then nC must satisfy Therefore assuming that N is sufficiently large so that not all passengers use the same transportation method the equilibrium allocation of passengers between the two transportation methods is given by Therefore Proposition 1710 The equilibrium number of passengers driving a car nC increases with the train fare φ and decreases with an increase in the value of time v Proposition 1710 is rather intuitive Clearly as the cost of operating the train hence the competitive fare rises more people will drive a car In addition since the train travel time is constant tT 1 as the value of time rises more people will use the train Page 454 1742 The socially optimal congestion level We now investigate from a social viewpoint what should be the optimal allocation of passengers We assume that the objective of the regulator is to minimize the aggregate time loss to passengers This measure is commonly used by regulators since it is assumed that loss of time to workers has a direct effect on the GNP Formally we define the regulators loss function by The regulator wishes to allocate the number of passengers on each mean of transportation to minimize L8 Formally the regulator solves Substituting N nC for nT the firstorder condition is given by Hence assuming an interior solution Consequently comparing 1722 with 1726 yields the following conclusions Proposition 1711 1 The socially optimal number of car users equals one half of the equilibrium number of car users 2 Subsidizing the train fare will reduce the number of car users but the ratio of the equilibrium car users to the optimal number of car users given by is independent of φ Part 2 of Proposition 1711 is of extreme importance since it has been observed that even in countries where public train systems are well developed traffic jams resulting from car congestion still prevail Thus the main message of Proposition 1711 is that traffic jams can be reduced but cannot be brought to the optimal level by providing cheap alternative transportation systems A diagrammatic illustration of the determination of the socially optimal and the equilibrium congestion levels is given in Figure 174 In Figure 174 the equilibrium number of car users is determined according Page 455 Figure 174 Equilibrium versus optimal highway congestion to condition 1721 which equates the loss from using the train to the total value of driving time In contrast the social planner allocates the passengers according to condition 1725 which equates the loss from using the train to the marginal value of driving time That is the social planner determines the number of car users so that the marginal change in loss from driving resulting from the addition of one more car driver equals the marginal and average loss from taking the train This explains why the social planner chooses a number of car users smaller than the equilibrium number 1743 Highway tolls We now show that the regulator should be able to reduce highway congestion to the optimal level by collecting a highway toll Let us suppose that the regulator collects a toll of τ dollars from each passenger who uses the highway What should be the exact toll that would bring the number of car users to the optimal level To solve this problem we calculate the equilibrium number of passengers when there is a toll From 1721 we have it that Hence when there is a toll the equilibrium number of car users is Equating ncτ to the optimal number of car users given in 1726 we have it that the optimal toll is given by Page 456 Therefore Proposition 1712 The optimal highway toll τ3 increases with the train fare φ and the value of time parameter v The intuition for Proposition 1712 is that an increase in the train fare would increase the number of car users Consequently there is a need for a higher toll to deter passengers from shifting from trains to private cars o 175 Exercises 1 Consider the costapproach airline model developed in subsection 1721 Suppose that the airlines cost function is given by For which values of α and β does the airline technology exhibit economies of scope according to Definition 171 assuming that all routes have equal numbers of passengers Prove your answer 2 Consider the demandapproach airline model developed in subsection 1722 a Suppose that the CAB has decided to regulate the airfare only on route 3 and to impose an airfare ceiling of where is given in 174 If the monopoly airline maintains the fully connected network show that the monopoly airline will reduce the frequency on route 3 That is calculate f3 Hint Calculus is not needed to answer this question b If the monopoly airline switches to the hubandspoke network and charges an airfare from route 3 passengers calculate f1 and f2 that would maximize the airlines profit and compare it to given in 174 Hint Reformulate the profitmaximization problem 175 taking into consideration that is a given constant 3 Consider the congestion model studied in section 174 but suppose now that the cardriving travel time is given by Perform the following exercises a Calculate the equilibrium number of passengers who drive their cars to work and the number of passengers who ride the train b Calculate the socially optimal allocation of passengers between private cars and public trains c If the socially optimal number of people who drive their cars to work is different than the equilibrium number find the optimal toll or subsidy that would implement the socially optimal number of car users Page 457 176 References Baumol W J Panzar and R Willig 1982 Contestable Markets and the Theory of Industry Structure New York Harcourt Brace Jovanovich Becker G 1974 A Theory of Social Interactions Journal of Political Economy 82 10631093 Becker G 1991 A Note on Restaurant Pricing and Other Examples of Social Influences on Price Journal of Political Economy 99 11091116 Berechman J and O Shy forthcoming The Structure of Airline Equilibrium Networks In Recent Advances in Spatial Equilibrium Methodologies and Applications A Volume in honor of T Takayama edited by J van den Bergh P Nijkamp and P Rietveld SpringerVerlag Bittlingmayer G 1990 Efficiency and Entry in a Simple Airline Network International Journal of Industrial Organization 8 245257 Borenstein S 1989 The Evolution of US Airline Competition Journal of Economic Perspectives 6 4573 Coase R 1950 The Problem of Social Cost Journal of Law and Economics 6 144 Conner K and R Rumelt 1991 Software Piracy An Analysis of Protection Strategies Management Science 37 125139 Cornes R and T Sandler 1983 On Commons and Tragedies American Economic Review 87 787792 Haverman R 1973 Common Property Congestion and Environmental Pollution Quarterly Journal of Economics 87 278287 Karni E and D Levin 1994 Social Attributes and Strategic Equilibrium A Restaurant Pricing Game Journal of Political Economy 102 822840 Levhari D and L Mirman 1980 The Great Fish War An Example Using a Dynamic Cournot Nash Solution Bell Journal of Economics 11 322334 Panzar J 1989 Technological Determinants of Firm and Industry Structure In Handbook of Industrial Organization edited by R Schmalensee and R Willig Amsterdam NorthHolland Posner R 1975 The Social Costs of Monopoly and Regulation Journal of Political Economy 83 807827 Sharkey W 1982 The Theory of Natural Monopoly Cambridge Cambridge University Press Viscusi K J Vernon and J Harrington 1992 Economics of Regulation and Antitrust Lexington Mass DC Heath and Company Weitzman M 1974 Free Access vs Private Ownership as Alternative Systems of Managing Common Property Journal of Economic Theory 8 225234 Page 459 INDEX 16 versus 13 A Action dominant 16 mixed 12 34 profile 13 34 pure 12 set 13 Address models see Location models Advertising 281 and concentration 298 and dealerships 383 and prices 299 and quality 297 and signaling 297 comparison 294 practice of 295 strategic use of 295 in usedgoods markets 377 informative 287 292 persuasive 283 292 DorfmanSteiner condition 284 regulations 300 targeted 290 experienced versus inexperienced consumers 291 Agency problem see PrincipalAgent problem Aircraft industry BoeingAirbus competition 241 light aircraft 377 Airline industry 440 hubandspoke networks 441 peakload pricing 349 350 352 Antitrust 5 Clayton Act 6 91 209 247 388 cooperative RD 247 development of 6 merger guidelines 210 per se rule 6 90 390 predatory pricing 89 212 price discrimination 91 price fixing 89 refusal to deal 89 212 RobinsonPatman Act 91 rule of reason 6 90 Sherman Act 5 6 71 89 90 212 389 trebledamage penalty 90 247 Arbitrage 75 76 346 international 369 370 Average cost 45 B Backward induction see Induction BainSylos postulate 187 188 192 Bertrand market structure 60 with capacity constraints 110 with differentiated products 139 Page 460 with homogeneous products 107 yielding Cournot outcome 112 Bestresponse function 21 36 100 138 190 193 243 409 411 425 strategic complements 140 strategic substitutes 140 243 Black box 221 Booms and recessions 120 Bundling 362 C Calculusfree topics xviii Cartel 78 117 CES constant elasticity of substitution see Utility Chamberlins tangency condition 146 Characteristics approach to product differentiation 135 Chicago School 4 CIF costinsurancefreight price 122 Civil Aeronautics Board CAB 446 Clayton Act see Antitrust Coases Conjecture 80 CobbDouglas function 54 Collusion cartel 78 RD 231 selfenforcing 115 via managerial compensation 412 Compatibility 254 downward 254 of components 270 oneway 267 partial 266 Competitive behavior 63 Competitive equilibrium 65 increasing returns to scale 66 Competitive market see Perfect competition Computer industry 264 software 264 piracy of 440 Concentration 1 and advertising see advertising measures 171 fourfirm 172 HerfindahlHirshman 173 210 211 Congestion 452 Consumer surplus 52 Cournot market structure 103 monopoly market structure 74 Contestable market structure 170 206 Contract 397 399 401 twopart tariff 382 Copyright 246 Cost function 45 72 sunk 170 183 207 Cournot market structure 60 and welfare 103 175 in international trade 121 owners control vs managers 408 with differentiated products 137 with heterogeneous firms 126 with homogeneous products 98 193 with variable number of sellers 101 Critical mass 258 Customs union see FTA Page 461 D Deadweight loss 68 74 235 Dealership 380 and advertising 383 territorial 385 Decision rule see Strategy Decreasing returns to scale DRS see Returns to scale Demand constant elasticity 49 function 49 inverse 49 50 linear 49 system 136 Differentiated products 133 discount stores 421 horizontal 149 310 353 maximum principle 314 measure of differentiation 136 minimum principle 152 quality 307 tying 372 vertical 310 353 370 Discount parameter 29 118 196 236 store 422 Downstream firm 176 Duality between cost and production functions 47 Dumping see International trade Duopoly 59 Durability see Quality Durablegoods monopoly see Monopoly E EC European Community advertising regulations 302 economic integration 369 freetrade agreement 123 Economies of scope 442 Edgeworth cycles see Bertrand market structure with capacity constraints Elastic demand 50 Elasticity discriminating monopoly 77 price 50 End user 380 price 382 Entry barriers 182 deterrence 186 hitandrun 206 Equilibrium general 143 in dominant actions 17 Nash 18 in mixed actions game 35 nonexistence of 20 34 40 refinement of 25 off path 32 119 path 32 subgame perfect 27 118 184 186 192 197 undercutproof 160 386 European Community see EC Expectation 400 number of customers 424 number of searches 431 of price 423 of price reduction 428 subjective 403 Experience goods 282 322 327 Externalities congestion 452 fishing 448 free rider 404 407 advertising 298 network 254 entertainment 438 tragedy of the commons 448 F Factors substitute 45 supporting 44 Page 462 Federal Trade Commission see FTC Firstmover advantage 188 Fishing industry 448 Fixed cost 46 Flow good 80 FOB freeonboard price 122 Folk Theorem 33 Foreclosure 179 by tying 366 Free entry 59 Free rider see Externalities FTA freetrade agreement see International trade FTC Federal Trade Commission 6 209 212 301 336 412 G Games 13 and coordination 17 Battle of the Sexes 17 255 extensive form 22 23 104 normalform representation 25 vertex vertices 23 history 28 information imperfect 12 38 perfect 12 38 set 38 node 23 normal form 13 outcome 13 extensive form game 24 mixedactions game 35 payoffs 13 mixedactions game 35 repeated game 29 PeaceWar 14 Pilot and the Terrorist 23 Prisoners Dilemma 15 33 repeated 28 60 finitely 29 infinitely 30 117 subgame 26 39 proper 26 tree 23 GATT General Agreement on Tariffs and Trade 91 122 General equilibrium see Equilibrium H Health insurance 326 Homogeneous products 64 measure of homogeneity 136 Hubandspoke see Airline industry I IATA International Air Transport Association 78 91 446 Imperfect competition 59 Incentive compatibility 417 Incentive constraint 399 401 403 Increasing returns to scale IRS see Returns to scale Individual rationality 417 Induction backward 27 30 104 Inelastic demand 50 Information asymmetric 322 327 333 402 416 symmetric 332 333 401 Information set see Games Innovation durability tradeoff 317 licensing 239 process 101 221 major 223 minor 223 patent 235 product 222 race 224 Page 463 subsidies 241 Installed base 262 Insurance 326 International Air Transport Association see IATA International trade differentiated products 147 dumping 120 FTA freetrade agreement 122 trade creation 126 trade diversion 126 homogeneous products 120 industrial organization and 7 market segmentation 369 PD subsidies 241 tariff 123 J Joint production 350 Judo economics 198 L Law of one price 421 LeaderFollower model see Sequentialmoves market structure Legal tender see Medium of exchange Lemons see Quality Liability product 335 Licensing innovation 239 Light bulbs industry 315 Limit pricing 187 cost signaling 202 Location models 149 310 352 Love for variety 143 264 M Managers compensation 407 408 413 Marginal cost 46 Marginal product 44 Marginal revenue 51 in cartel 79 in Cournot 99 in monopoly 72 Market segmentation 346 372 in international markets 369 Market structure 59 Matrix game representation 14 Medium of exchange 109 Merger 173 foreclosure 368 and tying 367 conglomerate 174 guidelines 210 horizontal 78 173 175 vertical 173 176 Money see Medium of exchange Monitoring 397 401 404 perfect 28 400 401 Monopolisticcompetition market structure 143 international trade and 147 Monopoly 59 71 discriminating 59 75 double markup 381 durable goods 80 multiplant 80 nondiscriminating 59 profit maximization in 72 renter 86 welfare in 73 Moral hazard 331 Multiplant monopoly see Monopoly Multiproduct firms 208 N Nash equilibrium NE see Equilibrium Network externalities see Externalities Newly developed material xxi Node see Games O Oligopoly 59 cooperative 59 Page 464 noncooperative 59 OPEC Organization of Petroleum Exporting Countries 91 P Pareto domination 22 117 efficient 22 noncomparable 22 optimal see Pareto efficient Participation constraint 399 400 403 Patent 224 233 history 245 law 244 optimal duration 233 types 245 Peakload pricing see Pricing Per se rule see Antitrust Perfect competition 59 63 increasing returns to scale 66 welfare 68 Perfect monitoring see Monitoring Predatory pricing 89 212 Prerequisites xvii Price discrimination 75 369 antitrust 91 nonuniform pricing 346 Price dispersion 421 Price fixing 89 vertical 389 Pricing nonlinear 362 nonuniform 346 peakload 348 efficient 351 endogenous seasons 352 twopart tariff 342 under dealership 382 Principalagent problem 396 402 Prisoners Dilemma see Games Probability subjective 402 Process innovation see Innovation Product differentiation see Differentiated products Production function 44 PTT Public Telephone and Telegraph see Telecommunication industry Public utility 416 Q Quality 307 advertising and see Advertising and income distribution 308 and market for lemons 322 durability 315 versus innovation tradeoff 317 usedgoods markets 376 signaling 327 Swans independence result 315 317 Quasilinear utility see Utility R RD 1 101 221 cooperative 229 231 legal approach 247 expected discovery date 228 noncooperative 231 race 224 international 241 spillover 230 subsidies 241 Raising a rivals cost 206 Randomizing actions 33 Rationing rule 108 Reaction function see Bestresponse function Refusal to deal 89 212 390 Regulation of congestion 455 of firms Page 465 fishing 451 price floor 446 under unknown cost 416 Renting 82 Repeated game see Games Resale price maintenance 383 Research and Development see RD Reservation price 111 strategy 429 Reservation utility see Utility Residual demand 83 114 Restaurant industry 438 Returns to scale 45 145 207 RD 230 Revealed profitability 106 142 Revelation truthful 417 Revenue function 51 Risk aversion 402 RobinsonPatman Act see Antitrust Rule of reason see antitrust S Search for lowest price 426 sequential 428 expected number of searches 431 recall 431 Search goods 282 322 Segmentation see Market segmentation Selling 82 Sequentialmoves market structure 104 141 Services peakload pricing of 352 supporting 263 tying 372 Sherman Act see Antitrust Shirking 397 401 404 Signaling cost 202 quality 327 using advertising 297 using warranties 334 workers productivity 327 SPE see Subgame perfect equilibrium Stackelberg see Sequentialmoves market structure Standardization 254 variety tradeoff 259 Strategic substitutes 243 Strategic complements 140 Strategic substitutes 140 Strategy 14 24 imperfect information 38 repeated game 28 set 14 29 TitforTat 33 trigger 31 Cournot duopoly 118 Structureconductperformance 2 Subcontracting horizontal 206 Subgame see Games Subgame perfect equilibrium see Equilibrium Supply function 65 Swans independence result see Quality T Takeitorleaveit 398 Takeover see Merger Tax fishing 451 highway tolls 455 specific 94 Team 404 Telecommunication industry 256 Page 466 PTT Public Telephone and Telegraph peakload pricing 352 Threat credible 197 incredible 26 194 Time inconsistency 407 TitforTat see Strategy Trademark 301 Tragedy of the commons see Externalities Transportation cost 149 international 121 Trebledamage penalty see Antitrust Tying 362 363 foreclosure 366 legal approach 388 mixed 364 product differentiation 372 U Undercutproof equilibrium see Equilibrium Undercutting 109 207 Unit cost 64 Unit elasticity 50 Upstream firm 176 Usedgoods markets cars 323 light aircraft 377 textbooks 376 Utility CES constant elasticity of substitution 143 love for variety 143 264 quasilinear 53 342 reservation 398 V Vertex see Games Vertical restraints 380 legal approach 389 W Warranties 330 commercial law 336 Welfare 68 in Cournot market structure 103 in monopoly market structure 73 in perfect competition 68 4 Consider the contestablemarkets market structure defined in section 85 Suppose that in the industry there is one incumbent firm and several potential competitors all having identical technologies summarized by the cost function TCqi 100 qi2 where qi is the output of firm i Solve for a contestablemarkets equilibrium assuming that the inverse aggregate demand facing the industry is given by p 60 4Qd A função de custo total para a firma i é dada por TCqi 100 qi2 Para encontrar o custo marginal CMg que é a taxa de variação do custo total em relação à quantidade calculamos a primeira derivada da função de custo total em relação a qi CMg dTCdqi d100 qi2dqi 2qi A demanda inversa é dada por p 60 4Q onde Q é a quantidade total do mercado No equilíbrio contestável o preço se iguala ao custo marginal No equilíbrio temos que CMg P Então 2qi 60 4qi Aqui estamos supondo que a quantidade total do mercado Q é igual à quantidade produzida pela firma incumbente qi pois no equilíbrio contestável uma firma potencial entrante produziria a mesma quantidade que a firma incumbente Agora vamos resolver a equação para qi 2qi 4qi 60 6qi 60 qi 606 qi 10 Com a quantidade de equilíbrio qi encontrada podemos agora calcular o preço de equilíbrio P substituindo qi de volta na função de demanda inversa P 60 4qi P 60 410 P 60 40 P 20 Finalmente vamos verificar se não há incentivos para a entrada ou saída de empresas Para isso calculamos o lucro total Π no equilíbrio onde o lucro total é a receita total menos o custo total Π P qi TCqi Π 20 10 100 10² Π 200 100 100 Π 200 200 Π 0 O lucro total é zero o que indica que não há incentivos para a entrada ou saída de empresas no mercado Portanto chegamos ao equilíbrio contestável do mercado com uma quantidade de equilíbrio de 10 unidades e um preço de equilíbrio de 20 unidades monetárias 910 Exercises 1 Consider the classification of process RD given in section 91 Suppose that the aggregate inversedemand function is given by p a Q and that initially all the firms have identical unit costs measured by c0 where c0 a 2c0 Suppose that one and only one of the firms is able to reduce its unit cost to c1 2c0 a Using Definition 91 infer whether this process innovation is considered to be minor or major Segundo a definição 91 temos duas condições para classificar uma inovação no processo de RD Inovação é considerada grande se pmc c0 Inovação é considerada pequena se pmc c0 As firmas inicialmente possuem custos idênticos c0 Uma firma consegue inovar reduzindo seu custo para c1 2c0 a Temos que c0 a 2c0 Precisamos verificar se c1 c0 ou c1 c0 Substituímos a na expressão c1 2c0 a Portanto c1 c0 o que significa que a inovação é grande segundo a definição 91 Esse raciocínio nos permite concluir que a inovação é considerada major pois permite à empresa inovadora um custo menor do que o custo unitário das firmas concorrentes alterando significativamente o potencial competitivo no mercado 2 Consider a threefirm version of the patentrace model studied in section 92 Suppose that each one of the three firms is capable of developing a product Let V denote the monetary value of the patent associated with the new product Each firm can construct a research lab provided that it invests I in the lab Assume that if a firm constructs a lab it has a probability of α 12 of discovering the product If only one firm discovers the product it will earn a profit equal to the full value of the patent ie V If only two firms discover then each will earn V2 and if all three discover then each will earn V3 Answer the following questions a Assuming that I1 calculate the minimal value of V that ensures that each firm will invest in constructing a lab b Suppose now that firm 3 went out of business and that a foreign firm purchased the two remaining firms Calculate the minimal value of V that would induce the foreign owner of the two firms to run the two separate research labs instead of operating only one lab Item a Cada firma tem a opção de construir um laboratório o que custa I Se decidir construir a firma tem uma probabilidade α de descobrir o produto No nosso caso α 12 e I 1 O lucro esperado de uma firma que constrói um laboratório e descobre o produto sozinha é o valor do patente V menos o custo de investimento I Como a probabilidade de descobrir o produto é α o lucro esperado Eπ é Eπ αV I Para incentivar a firma a investir na construção do laboratório o lucro esperado deve ser não negativo Eπ αV I 0 Substituímos α por 12 e I por 1 12 V 1 0 Resolvendo a desigualdade para V obtemos V 2 O valor mínimo de V que garante que cada firma investirá na construção de um laboratório é 2 Isso significa que a firma está disposta a investir na construção de um laboratório se espera que o valor do patente seja pelo menos 2 Item b Com a saída da firma 3 e a aquisição das duas firmas restantes por um proprietário estrangeiro o cenário mudou Agora o proprietário estrangeiro deve decidir se opera um ou dois laboratórios O custo para operar um laboratório é I e para operar dois laboratórios é 2I Cada laboratório tem uma probabilidade α de descobrir o produto No nosso caso α 12 e I 1 Item a Item b Item c Item a Item b Item c Para dissuadir a Airbus de desenvolver o avião o subsídio dos EUA para a Boeing teria que ser tão alto que garantisse que a Boeing sempre teria um lucro maior do que a Airbus não importando a decisão de produção desta última Se a Airbus tem um lucro de 65 sem concorrência e 5 com concorrência o subsídio para a Boeing teria que ser maior do que 65 para que a Airbus sempre tivesse um lucro menor independentemente de suas ações Item d Esta é uma questão de análise econômica mais ampla Se ambos os governos entram numa guerra de subsídios podese argumentar que isso é ineficiente do ponto de vista global pois leva a um gasto excessivo de recursos públicos que poderiam ser melhor utilizados em outros lugares No entanto também pode haver argumentos a favor dos subsídios como o desenvolvimento de tecnologias avançadas a criação de empregos e a preservação da competitividade internacional A conclusão dependerá de uma ponderação desses vários fatores 104 Exercises 1 Consider the supportingservices approach model developed in subsection 102 a For a given hardware price of brand A pA what is the price of computer B beyond which firm B would have a zero market share b Suppose that pA pB and suppose that the income of each consumer doubles to 2Y while hardware prices remain unchanged Calculate the effect this increase in incomes on i the market shares δA and δB and on ii the ratio of the number of software packages written for computer A to the number of software packages written for computer B Item a A questão a pede para encontrarmos o preço de computador B pB tal que a participação de mercado da firma B seja zero ou seja δB 0 Para isso vamos utilizar a equação 109 do livro que relaciona a participação de mercado de B δB com a de A δA e a quantidade de software disponível para cada marca NA e NB NBNA δ1δ E a participação de mercado de B δB é dada por δB 1 δ Quando δB é 0 temos δA 1 100 do mercado prefere a marca A Se substituirmos δA 1 na equação acima temos que a razão NBNA será 0 o que implica que NB também deve ser 0 porque a quantidade de software para B NB seria 0 se ninguém estiver usando a marca B Agora usando a equação 1010 do livro temos NB 1 δ Y pB Como queremos que NB seja 0 para que δB seja 0 substituímos NB por 0 e δ por 1 o que nos dá 0 1 1 Y pB Simplificando a equação 0 0 Y pB 0 0 Isso resulta em uma identidade indicando que com base nessas equações não há um número específico para pB que satisfaça essa condição já que qualquer valor para pB resultará em 0 0 uma vez que o termo 1 δ é zero Isto implica que para o modelo dado a questão de encontrar um pB tal que δB 0 não tem uma solução única No contexto do modelo isso pode significar que sempre haverá alguma participação de mercado para a firma B independentemente do seu preço desde que as outras condições do modelo se mantenham Item b Para resolver o item b do exercício passo a passo temos a seguinte situação o preço do computador A pA é maior que o preço do computador B pB e a renda de cada consumidor dobra para 2Y enquanto os preços dos hardwares permanecem inalterados Precisamos calcular o efeito desse aumento na renda nas participações de mercado δA e δB e na razão do número de pacotes de software escritos para o computador A em relação ao B Começaremos com a equação 1010 que nos fornece a expressão para δA e implicitamente δB já que δB 1 δA δA EAEA EB δB EBEA EB onde EA e EB são os gastos totais dos consumidores nos softwares compatíveis com os computadores A e B respectivamente Quando a renda do consumidor dobra de Y para 2Y podemos substituir Y por 2Y nas expressões para EA e EB e obter as novas expressões para δA e δB Segundo a equação 1011 do livro temos EA δA 2Y pA EB δB 2Y pB Substituímos EA e EB nas expressões para δA e δB δA δA 2Y pAδA 2Y pA δB 2Y pB δB δB 2Y pBδA 2Y pA δB 2Y pB Como δA δB 1 podemos resolver essas equações para δA e δB em termos de 2Y pA e pB Agora vamos calcular o efeito sobre a variedade de software disponível para cada marca de computador usando a equação 108 1 δA NA δB NB NA δA 2Y pA NB δB 2Y pB Com a renda dobrada essas expressões para NA e NB também dobrarão pois são diretamente proporcionais à renda 2Y A razão do número de pacotes de software para A e B após o aumento da renda será então Razão após o aumento da renda NBNA δB 2Y pBδA 2Y pA Para resolver essa equação passo a passo precisaríamos dos valores específicos de pA pB e Y No entanto podemos dizer que a razão entre o número de pacotes de software para A e B após o aumento da renda será determinada pela proporção dos gastos dos consumidores entre os softwares A e B que agora são baseados em uma renda dobrada Se os preços dos computadores permanecerem inalterados o aumento da renda aumentará os gastos em ambos os softwares mas a proporção de gastos e portanto a proporção da variedade de software pode mudar dependendo de como pA e pB se comparam um com o outro Se pA for significativamente maior que pB poderíamos esperar que a variedade de software para B aumentasse mais que para A já que os consumidores teriam mais renda disponível para gastar com B o que é mais acessível Caso contrário se os preços forem próximos a mudança na variedade de software poderia ser mais uniformemente distribuída entre A e B 2 Consider the component approach analyzed in subsection 103 but assume that there are four consumers consumer AA consumer BB consumer AB and consumer BA a If the components are incompatible prove that no NashBertrand equilibrium in system prices pA and pB as defined in Definition 105 exists b If the components are compatible calculate the symmetric equilibrium prices of all components firms profit levels and consumers surplus Item a Vamos resolver passo a passo o item a da questão 2 que lida com a incompatibilidade dos componentes e a demonstração de que não existe um equilíbrio de NashBertrand para os preços dos sistemas pA e pB Com componentes incompatíveis temos dois sistemas distintos XA YA e XB YB e cada sistema só pode ser vendido a consumidores que preferem especificamente aquele sistema Os consumidores são AA BB AB e BA Cada firma tem um preço para o seu sistema completo que é a soma dos preços dos componentes individuais De acordo com a Proposição 1013 três equilíbrios são possíveis com componentes incompatíveis A firma A vende para AA e AB a firma B vende para BB A firma B vende para BB e AB a firma A vende para AA A firma A vende para AA a firma B vende para BB AB não é atendido Para que um equilíbrio de NashBertrand exista cada firma deve estabelecer um preço tal que nenhuma delas possa aumentar seu lucro unilateralmente mudando seu próprio preço No contexto de componentes incompatíveis isso significa que cada firma deveria ser incapaz de reduzir seu preço para capturar os consumidores da outra firma sem que isso resulte em perda de lucro Para provar que não existe um equilíbrio de NashBertrand precisamos mostrar que uma firma pode sempre aumentar seu lucro reduzindo o preço o que contraria a definição de equilíbrio Por exemplo se a firma A reduzisse seu preço ligeiramente abaixo do preço da firma B ela capturaria o consumidor BB além de AA e AB aumentando seu lucro Isso continuaria até que o preço caísse a um nível que não cobrisse mais os custos o que não é sustentável A prova está na possibilidade de cada firma poder sempre desviarse dos preços estabelecidos para capturar todos os consumidores o que indica que não pode existir um preço estável onde nenhuma firma tenha o incentivo para desviar Portanto concluímos que não existe um equilíbrio de NashBertrand nos preços dos sistemas quando os componentes são incompatíveis Essa conclusão é reforçada pelo fato de que em um mercado onde os componentes são incompatíveis e o consumidor AB não é atendido as firmas não encontram um ponto de preço estável onde ambas concordem em não reduzir os preços já que cada firma tem o incentivo para reduzir seu preço e capturar o consumidor da outra o que é uma característica de mercados altamente competitivos e uma indicação da inexistência de um equilíbrio de NashBertrand Item b Quando os componentes são compatíveis segundo a Definição 106 um equilíbrio de componentes compatíveis é um conjunto de preços e quantidades para os componentes vendidos por cada firma de modo que dados os preços cada firma escolhe os preços para maximizar o número de consumidores que escolhem seus componentes levando em consideração a preferência dos consumidores De acordo com a Proposição 1014 existe um equilíbrio onde cada consumidor compra seu sistema ideal e todos os componentes são igualmente precificados em λ e os níveis de lucro das firmas são πA πB 3λ Cada firma vende dois componentes de cada tipo Como os preços estão em equilíbrio em λ e sabendo que cada firma vende quatro componentes no total dois de cada o lucro de cada firma é de 4λ custos No entanto não temos informações sobre os custos então assumimos que os lucros são dados diretamente pela Proposição 1014 O excedente do consumidor é definido como a soma das utilidades dos consumidores Como os consumidores estão pagando exatamente o que valorizam pelos sistemas seus preços de reserva são iguais aos preços de equilíbrio o excedente do consumidor é zero pois eles não obtêm utilidade adicional além do que pagam O bemestar social é definido como a soma dos níveis de lucro das firmas e do excedente do consumidor Já que o excedente do consumidor é zero e os lucros das firmas são 3λ cada o bemestar social é simplesmente a soma dos lucros das duas firmas ou seja 6λ Portanto em equilíbrio com componentes compatíveis temos Preços de Equilíbrio dos Componentes λ para cada componente tanto para A quanto para B Lucros das Firmas 3λ para cada firma totalizando 6λ para ambas Excedente do Consumidor 0 pois os consumidores estão pagando exatamente o que seus sistemas valem para eles BemEstar Social 6λ o que é a soma dos lucros das firmas já que o excedente do consumidor é zero Esses resultados implicam que em um mercado com componentes compatíveis os preços tendem a ser uniformizados e os consumidores compram os sistemas que correspondem exatamente à sua valoração resultando em um excedente do consumidor nulo e um bemestar social que é composto exclusivamente pelos lucros das firmas Item a A função de demanda é dada por QApβAλA pεp Onde β é o parâmetro de efetividade da publicidade λA é a elasticidade da publicidade εp é a elasticidade do preço da demanda A é o nível de publicidade p é o preço do produto A receita do monopolista é πApp QAp c QAp A p βAλA pεp c βA λA pεp A As condições de primeira ordem para o preço p e o nível de publicidade A são respectivamente πApp0 πApA0 A condição de primeira ordem para o preço é dada por 0βAλA εp 1 pεp c β AλA εp pεp 1 Da qual isolamos p para encontrar o preço ótimo pM εpεp1c A condição de primeira ordem para o nível de publicidade é 0βλA AλA 1 pεp 1 1 Da qual isolamos A para encontrar o nível ótimo de publicidade AM β λA pεp 111λ A 1 De acordo com o exercício temos εA005 e εp 02 e o custo unitário c1 Substituímos esses valores na equação de AM para encontrar o nível ótimo de publicidade 117 Exercises 1 Congratulations You have been appointed to become a CEO of UGLY Inc the sole producer of facial oil skinlife extender Your first assignment is to determine the advertising budget for next year The marketing department provides you with three important information items a The company is expected to sell 10 million worth of the product b It is estimated that a 1 percent increase in the advertising budget would increase quantity sold by 005 percent c It is also estimated that a 1 percent increase in the products price would reduce quantity sold by 02 percent a How much money would you allocate for advertising next year b Now suppose that the marketing department has revised its estimation regarding the demand price elasticity to 1 percent increase in price resulting in a reduction in quantity sold by 05 percent How much money would you allocate to advertising after getting the revised estimate c Conclude how a change in the demand price elasticity affects advertising expenditure Os cálculos resultaram nos seguintes níveis ótimos de publicidade Para a elasticidade de preço original um aumento de 1 no preço reduz a quantidade vendida em 02 o nível ótimo de publicidade é 025 x p5M Item b Para o item b vamos calcular o orçamento de publicidade após a revisão da elasticidadepreço da demanda A nova elasticidadepreço da demanda é de 05 o que significa que um aumento de 1 no preço leva a uma redução de 05 na quantidade vendida A nova elasticidadepreço da demanda é εp 05 Usando a condição de primeira ordem para o preço atualizamos o preço ótimo do monopolista pM para refletir a nova elasticidadepreço pM εp εp 1 c A condição de primeira ordem para o nível de publicidade Equação 117 com a nova elasticidadepreço da demanda é 0 β Aλ λ 1 pεp 1 1 Resolvemos esta equação para A para encontrar o novo nível ótimo de publicidade AM Após a revisão da elasticidade de preço um aumento de 1 no preço agora reduz a quantidade vendida em 05 o nível ótimo de publicidade permanece o mesmo 025 x p5M 2 In Future City there are two fortunetellers Ms α and Mr β Each fortuneteller charges a fixed regulated fee of 10 for one visit Let Ai denote the advertising expenditure of fortuneteller i i α β The number of clients visiting each teller per unit of time is denoted by ni i α β We assume that ni depends only on the advertising expenditure of both tellers Formally let nα 6 3 Aβ Aα and nβ 6 3 Aα Aβ Thus the number of clients visiting teller α increases with αs advertising expenditure and decreases with βs advertising expenditure Altogether assume that each fortuneteller i has only one choice variable which is the advertising level and therefore chooses Ai to maximize the profit given by πi Aα Aβ 10 ni Ai i α β a Compare the number of visitors and the profit level of each fortuneteller when Aα Aβ 1 and for Aα Aβ 2 What can you conclude about the role of advertising in this city b Calculate and draw the bestresponse function of teller β as a function of the advertising expenditure of teller α In case you forgot how to define bestresponse functions we first used them in section 61 c Calculate the tellers advertising level in a Nash equilibrium d In view of your answer to a is the Nash equilibrium you found in c optimal for the fortuneteller industry e Is the equilibrium you found stable Item a A função de lucro de cada adivinho é dada por πAα Aβ 10 ni Ai onde ni é o número de visitantes e Ai é o gasto com publicidade O número de visitantes para cada adivinho é dado por nα 6 3 Aβ Aα nβ 6 3 Aα Aβ Substituímos Aα e Aβ por 1 na função de lucro πAα Aβ 106 3 1 πAα Aβ 103 1 πAα Aβ 30 1 πAα Aβ 29 Agora substituímos Aα e Aβ por 2 πAα Aβ 10 6 322 2 πAα Aβ 103 2 πAα Aβ 30 2 πAα Aβ 28 Isso indica que aumentar os gastos com publicidade acima de 1 para cada adivinho não é benéfico pois reduz o lucro total Em outras palavras o gasto adicional com publicidade não gera receita adicional suficiente para cobrir o custo da publicidade extra o que sugere que há um ponto de saturação no impacto da publicidade sobre a atração de novos clientes Item b A função de lucro do adivinho β com base na quantidade de visitantes que depende dos gastos com publicidade de ambos os adivinhos é πβAα Aβ 10 nβ Aβ πβAα Aβ 106 3Aα Aβ Aβ Para encontrar a função de resposta ótima derivamos a função de lucro em relação ao gasto com publicidade de β e igualamos a zero dπβdAβ 30 Aα Aβ2 1 Resolvemos a derivada para Aβ para encontrar a função de resposta ótima 0 30 Aα Aβ2 1 Aβ2 30 Aα 1 Como estamos lidando com gastos reais e positivos a resposta ótima de β será a raiz quadrada positiva do valor absoluto do lado direito da equação A função de resposta ótima do adivinho β em função do gasto com publicidade do adivinho α é 5477 x sqrtAα Para o equilíbrio de Nash onde ambos os adivinhos escolhem o mesmo nível de publicidade encontramos que o adivinho α deve gastar 30 em publicidade para que sua escolha seja ótima Dado que a função de resposta ótima de β é proporcional à raiz quadrada do gasto de α podemos inferir que β também escolherá um nível de publicidade que seja uma função da raiz quadrada do gasto ótimo de α Item c No equilíbrio de Nash ambos os adivinhos escolhem gastar 30 em publicidade Isso significa que no equilíbrio cada adivinho ajusta seu nível de publicidade para maximizar seu próprio lucro considerando o nível de publicidade do outro adivinho No contexto deste problema isso resulta em um gasto de 30 em publicidade por cada adivinho Item d Encontramos que no equilíbrio de Nash ambos os adivinhos gastam 30 em publicidade No item a determinamos que quando ambos os adivinhos gastam 1 em publicidade o lucro de cada um é 29 enquanto que ao aumentar o gasto para 2 o lucro cai para 28 Isso sugere que aumentos no gasto com publicidade além de 1 levam a reduções no lucro O aumento dos gastos com publicidade de 1 para 30 no equilíbrio de Nash não aumentou os lucros Na verdade com base nas conclusões do item a o lucro poderia diminuir com o aumento da publicidade uma vez que a receita adicional gerada por visitantes extras pode não compensar o aumento nos custos de publicidade A conclusão é que o equilíbrio de Nash encontrado pode não ser ótimo para a indústria de adivinhos Se a publicidade serve apenas para redistribuir os clientes existentes então os custos adicionais de publicidade não aumentam o número total de clientes resultando em uma guerra de despesas que diminui o lucro global da indústria A otimização do lucro da indústria como um todo poderia envolver uma estratégia de publicidade coordenada ou limitada que evite a concorrência de soma zero onde os gastos aumentam sem aumentar o tamanho do mercado Item e No contexto da questão fornecida o equilíbrio de Nash é estável no sentido de que nenhum dos adivinhos tem um incentivo unilateral para desviar do seu nível de publicidade escolhido dado que o outro adivinho está fazendo a mesma coisa Em outras palavras cada adivinho está respondendo otimamente aos níveis de publicidade do outro e nenhuma das partes pode melhorar seu lucro mudando apenas o seu próprio nível de publicidade No entanto a estabilidade do equilíbrio de Nash não implica que ele seja socialmente ótimo Embora os adivinhos não tenham incentivo para mudar suas estratégias individualmente o equilíbrio resulta em um gasto excessivo em publicidade que não aumenta o tamanho do mercado mas apenas redistribui os clientes existentes Isso leva a uma ineficiência do ponto de vista da indústria como um todo onde o custo combinado de publicidade pode superar o benefício obtido por meio do aumento da participação de mercado 3 Prove part 4 of Proposition 115 Hint Follow the same steps as in the proof of part 3 A firma 1 usa publicidade informativa I A firma 2 usa publicidade persuasiva P IP é um equilíbrio se θ max NE 1 NE onde θ é a popularidade da marca entre usuários experientes N é o lucro quando ambas as firmas não desviam da estratégia e E é o lucro quando ambas desviam para publicidade informativa A firma 1 não desviará de sua estratégia de publicidade informativa se o lucro obtido por essa estratégia for maior do que o lucro obtido por desviar para publicidade persuasiva Se θ é alto o suficiente isso indica que a marca já é bem estabelecida entre os usuários experientes fazendo com que a publicidade informativa seja mais eficaz e a firma não tenha incentivo para desviar para a publicidade persuasiva A firma 2 usará publicidade persuasiva e não terá incentivo para desviar para informativa se a eficácia da publicidade persuasiva aumentar a popularidade entre usuários inexperientes for maior que o benefício de desviar para informativa Novamente se θ for alto a eficácia da publicidade informativa da Firma 1 é reforçada e a Firma 2 não pode lucrar mais com a mudança para publicidade informativa Para I P ser um equilíbrio o valor de θ deve ser tal que nenhuma das firmas tenha incentivo para desviar das suas estratégias de publicidade o que é garantido se θ for maior que ambos NE e 1 NE Isso ocorre porque um alto θ significa que a marca é suficientemente popular entre os usuários experientes então a Firma 1 pode se concentrar em informar sobre o produto enquanto a Firma 2 se concentra em persuadir novos usuários 129 Exercises 1 Consider the modified Hotelling verticaldifferentiation model of subsection 1222 but suppose that consumers have a reservation utility in the sense that a consumer prefers not to buy any brand if his or her utility falls below zero Recall that the preferences exhibited in 122 imply that there is no lower bound on utility from consumption Figure 123 implies that this modification in preferences would not affect the number of highqualitybrand buyers since all consumers indexed on 1 gain a strictly positive utility from buying the highquality brand However point z in Figure 123 shows that no consumers indexed on 0 z will purchase any brand since otherwise their utility falls below zero Perform the following exercises a Show that for given a b pA and pB the number of consumers who do not purchase any brand equals to z pAa Para começar precisamos entender o contexto do modelo de diferenciação vertical de Hotelling modificado para incluir a utilidade de reserva No modelo um consumidor só comprará uma marca se a utilidade obtida dessa marca for maior que zero A utilidade de reserva é o ponto abaixo do qual o consumidor prefere não comprar nenhum produto Dado isso a utilidade para um consumidor de índice i para as marcas A e B é dada por UAi ai PA e UBi bi PB respectivamente O ponto z no modelo é o ponto onde a utilidade de comprar uma marca cai para zero Nesse caso a utilidade de reserva é zero e qualquer utilidade abaixo disso resultará na decisão de não comprar Assim para encontrar z igualamos a utilidade de comprar a marca A a zero já que estamos procurando o ponto onde os consumidores não compram nenhuma marca 0 az PA Resolvendo para z temos Isso mostra que o número de consumidores que não compram nenhuma marca é z PAa b Conclude that the market share of firm A is x z PB PAb a PAa Dado que x PB PAb a pela equação 124 e z PAa conforme encontrado na parte a do exercício podemos substituir esses valores na expressão para encontrar a participação de mercado da empresa A Substituindo x e z na expressão temos x z PB PAb a PAa c Using the same procedure as in 125 show that for given a and b the secondperiod equilibrium prices and profit levels are given by pAab aba 4b a and pBab 2bba 4b a πAab abba 4b a2 and πBab 4b2ba 4b a2 As fórmulas de equilíbrio para os preços e níveis de lucro no segundo período dados a e b são pAeab aba 4ba pBeab 2bba 4ba πAeab abba 4ba2 πBeab 4b2ba 4ba2 O lucro de cada empresa é a área do retângulo formado pela quantidade de unidades vendidas que é a participação de mercado e o preço acima do custo marginal que para este modelo assumimos ser zero para simplificar a análise Primeiro vamos encontrar as expressões de maximização de lucro para ambas as empresas O lucro da empresa A πA e da empresa B πB são dados por πA PA x πB PB 1 x Onde x é a participação de mercado da empresa A que já foi encontrada anteriormente como PB PA b a Para maximizar o lucro de cada empresa tomamos a derivada do lucro em relação ao preço de cada empresa e igualamos a zero Vamos calcular essas derivadas e resolver as equações para obter os preços de equilíbrio As expressões simplificadas dos preços de equilíbrio usando o procedimento de maximização do lucro são pAeab a 3 b 3 pBeab 2a 3 2b 3 Essas expressões diferem das fornecidas na questão Parece haver um equívoco Isso pode ocorrer devido à especificidade do modelo em questão que pode incluir outras variáveis ou restrições não detalhadas nas equações que temos Para continuar e resolver corretamente precisaríamos entender completamente a função de lucro que está sendo maximizada e quaisquer outras condições ou restrições impostas no modelo d Show that in the first period firm A would choose to locate at ae 47 whereas firm B would locate at be 1 No modelo de Hotelling clássico as empresas tendem a se localizar no ponto médio do espaço de produto para capturar a maior parte do mercado possível supondo uma distribuição uniforme dos consumidores ao longo desse espaço Contudo este problema parece sugerir uma variação desse modelo onde as empresas têm localizações ótimas específicas ae e be que não são necessariamente no ponto médio Para mostrar que ae 47 e be 1 precisaríamos entender a função de utilidade dos consumidores a distribuição dos consumidores no espaço de produto e como as empresas competem em termos de localização Tipicamente resolveríamos isso configurando e resolvendo um modelo que inclui esses elementos No entanto sem informações adicionais ou equações relevantes não podemos proceder diretamente 2 Prove the second part of Proposition 121 using the same procedure as the one used in the proof of the first part Dado que U1H e U2H representam a utilidade de comprar a marca de alta qualidade para o consumidor de alta renda 1 e baixa renda 2 respectivamente U1L e U2L representam a utilidade de comprar a marca de baixa qualidade para o consumidor de alta renda 1 e baixa renda 2 respectivamente pH e pL são os preços das marcas de alta e baixa qualidade respectivamente H e L representam a renda dos consumidores de alta e baixa renda respectivamente com H L Podemos expressar a utilidade de comprar cada marca para cada consumidor como U1H H pH U1L H pL U2H L pH U2L L pL Agora vamos supor que o consumidor de alta renda escolha a marca de baixa qualidade Isso significa que U1L U1H ou seja H pL H pH Simplificando obtemos pH pL Agora precisamos mostrar que U2L U2H também é verdade ou seja que L pL L pH Como sabemos que pH pL então a desigualdade é verdadeira 3 Consider the lemon model described in section 125 and suppose that the owner of the good used car must sell his or her car because he or she is leaving the country Assume that the market prices of used and new cars are exogenously given by 0 pU UG 2 and pN NG 2 respectively Characterize the demand and supply patterns of the four types of agents under these prices A estrutura clássica do modelo de leilão de carros usados envolve os seguintes agentes Vendedores de carros usados de boa qualidade Good used cars Vendedores de carros usados de má qualidade Lemons Compradores de carros usados Compradores de carros novos Os preços de mercado são dados como pU o preço de um carro usado onde 0 pU UG 2 sendo UG a utilidade de um carro novo para um comprador que adquire um carro usado de boa qualidade pN NG 2 o preço de um carro novo onde NG é a utilidade de um carro novo Os padrões de demanda e oferta para cada tipo de agente sob esses preços são determinados pelas suas utilidades esperadas e pelos preços de mercado Vamos caracterizar esses padrões com base nas informações fornecidas Vendedores de carros usados de boa qualidade Good used cars Estes vendedores têm uma utilidade UG de um carro novo e irão vender seus carros se a utilidade esperada de um carro novo for maior do que a utilidade esperada de manter o carro usado mais o preço de venda do carro usado Vendedores de carros usados de má qualidade Lemons Estes vendedores têm uma utilidade UL de um carro novo que é menor que UG e irão vender seus carros se puderem obter um preço que compense a perda de utilidade de não ter um carro Compradores de carros usados Eles irão comprar um carro usado se a utilidade esperada de um carro usado for maior do que o preço de mercado do carro usado Compradores de carros novos Eles irão comprar um carro novo se a utilidade de um carro novo for maior do que o preço de mercado do carro novo Os agentes farão suas decisões baseadas no cálculo de suas utilidades esperadas que dependem das probabilidades de um carro usado ser de boa ou má qualidade e compararão essas utilidades esperadas com os preços de mercado A presença de assimetria de informação onde os vendedores conhecem a qualidade de seus carros mas os compradores não leva ao problema clássico do mercado de limões descrito por Akerlof onde a qualidade média dos carros no mercado pode declinar e os preços de mercado podem não refletir adequadamente a qualidade dos carros Para caracterizar precisamente os padrões de demanda e oferta precisaríamos calcular as utilidades esperadas para cada tipo de agente e comparálas com os preços de mercado dados Isso geralmente envolve a criação de equações baseadas nas funções de utilidade e na resolução dessas equações para encontrar equilíbrios de mercado Os padrões de demanda e oferta para os quatro tipos de agentes sob os preços dados são caracterizados pelas seguintes condições Compradores de carros usados Eles irão comprar um carro usado se a utilidade esperada de um carro usado de boa qualidade UG pU for maior que zero e se a utilidade esperada de um carro usado de má qualidade UL pU também for maior que zero Vendedores de carros usados de boa qualidade Good used cars Como o proprietário do carro usado de boa qualidade precisa vender o carro pois está deixando o país podemos assumir que ele irá fornecer ao mercado desde que o preço pU seja maior que zero Vendedores de carros usados de má qualidade Lemons Eles também irão fornecer ao mercado pois o preço pU é maior que zero o que sugere que eles têm incentivo para vender Compradores de carros novos Eles irão comprar um carro novo se a utilidade de um carro novo NC2 for maior que zero Essas condições são baseadas nas informações fornecidas e assumem que a utilidade de possuir um carro tanto novo quanto usado é positiva para os compradores Para os vendedores a condição é que eles estão dispostos a vender a qualquer preço positivo especialmente porque o vendedor do carro usado de boa qualidade está saindo do país b What is the maximum price the monopoly can charge for the product sold with this type of warranty O custo esperado para o monopólio como calculamos anteriormente é c 2 p O preço máximo que o monopólio pode cobrar pelo produto será o valor que o consumidor espera do produto menos o custo esperado que o monopólio tem ao fornecer a garantia Isso é porque o consumidor estará disposto a pagar até o valor que ele atribui ao produto ajustado pela probabilidade de ter que usar a garantia A fórmula para o preço máximo é Preço máximo V Custo esperado da garantia Preço máximo V c 2 p Esta é a expressão para o preço máximo que o monopólio pode cobrar pelo produto sob a nova política de garantia c Conclude whether Proposition 1211 holds for this type of warranty O preço máximo que o monopólio pode cobrar pelo produto com essa garantia limitada é V c 2 p conforme calculado anteriormente Para que valha a pena oferecer a garantia o preço cobrado com a garantia limitada deve ser maior do que o custo esperado para o monopólio o que implica que V c 2 p c ou de outra forma V c 3 p Se essa condição for satisfeita então o monopólio ainda terá um incentivo para oferecer a garantia limitada já que ele pode cobrar um preço que excede o custo esperado de fornecer a garantia mesmo considerando que a garantia só cobre uma substituição